USIT-405-Computer-Graphics-and-Animation-munotes

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UNIT I
1

INTRODUCTION TO COMPUTER
GRAPHICS

Unit Structure
1.0 Objectives
1.1 Overview of Computer Graphics
1.1.1 Definitions
1.1.2 Advantages of Computer Graphics
1.2 Computer Graphics Applications
1.2.1 Computer Aided Design
1.2.2 Computer Art
1.2.3 Entertainment
1.2.4 Education and Training
1.2.5 Presentation Graphics
1.2.6 Visualization
1.2.7 Image Processing
1.2.8 Graphical User Interfaces
1.3 Computer Graphics Software
1.3.1 Coordinate Representations
1.3.2 Graphic Functions
1.3.3 Software Standards
1.4 Input Devices
1.4.1 Keyboards
1.4.2 Mouse
1.4.3 Trackball and Spaceball
1.4.4 Joysticks
1.4.5 Data Glove
1.4.6 Image Scanner
1.4.7 Touch Panels
1.4.8 Light Pens
1.4.9 Voice Systems
1.5 Active and Passive Graphics
1.6 Summary
1.7 Unit End Exercise
1.8 References
1.9 Bibliography munotes.in

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1.0 OBJECTIVES
After this chapter, student will be able to:
 Define computer graphics
 State various applications of graphics
 Understand coordinate system and various graphics functions
 Identify various input devices used in graphics

1.1 OVERVIEW OF COMPUTER GRAPHICS
 Computer graphics deals with representation of information in pictorial
form.
 It is a way of generating of images or any kind of object with the aid of
computers.
 It plays an important role in many fields like education, photography,
filming and many more.

1.1.1 Definitions :
 Computer Graphics : It is a field of computer science that deals with
creating, manipulating, storing and displaying pictures or objects on a
display device.
 Pixel : It is a smallest element on the display screen .
 Resolution : The number of pixels on a display monitor expressed in
terms of number of pixels on horizontal axis and number of pixels on
vertical axis is called as resolution.
 Rasterization : The process of determining appropriate pixels for
representing pictu re or obj ect on screen is called as rasterization.
 Scan conversion : The process of representing continuous picture or
object as a collection of discrete pixel is called as scan conversion.

1.1.2 Advantages o f Computer Graphics:
 Produces high quality graphics.
 Provides use of tools to generate and manipulate objects or pictures.
 Used to work with 2D and 3D objects.
 Used in animations.

1.2 COMPUTER GRAPHICS APPLICATIONS
 Computer graphics ca n be used in various fields. Some of the common
applications are Computer Aided Design, Computer Art, Entertainment
and Education and Training
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1.2.1 Computer Aided Design :
It is used in
 Design processes of architecture and engineering systems .
 Design of automobiles , textiles, spacecraft, aircrafts, buildings, etc.


Fig. 1 Use of CAD in computer graphics

1.2.2 Computer Art :
 It is majorly used in the field of fine arts and commercial applications.
 Use of applications like mathematics package, paint package, animation
program and desktop publishing software helps in computer art . E.g.,
cartoon drawing, painting.


Fig. 2 Use of Computer Art in computer graphics

1.2.3 Entertainment :
 It can be used to create any motion pictures and videos.
 Animated movies or video clips can be created.
 Sound and music can be used to generate any movie clips, animated
shows or TV shows.
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 Games can also be made with interactivity features.

Fig. 3 Use of Entertainment in computer graphics

1.2.4 Education and Training :
 It is used in classroom to enhance educational experience by
simplifying teacher’s work.
 Models can be generated which can be useful for teaching and making
understanding more easier.
 In distance learning, where students are not at the same place, use of
computer graphics plays an important role, by means of conducting
tests. E.g. , kahoot.it, quizzes.com.
 For trai ning, simulators can be made as hands on experience for
candidates for better understanding .


Fig. 4 Use of Kahoot in Education

1.2.5 Presentation Graphics :
 Reports can be generated on slides.
 It is used for summarizing mathematical, financial, statistical and
scientific data.
 Graphs and charts can be generated to make any kind of presentation.
E.g., pie charts , bar charts.
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Fig. 5 Use of Presentation graphics in computer graphics

1.2.6 Visualization :
 To check insights of data, to study behavior of processes, visualization
is required with proper use of computer graphics.



Fig. 6 Use of Visualization in computer graphics

1.2.7 Image Processing :
 Used in applications where images or photographs requires editing.
 Processing on existing images done for more enhancement s.




Fig. 7 Use of Image Processing in computer graphics

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1.2.8 Graphical User Interfaces :
 A window manager is an important component in GUI that allows a
user to display multimedia window areas.
 For fast selection processing, use of menus, icons and buttons are made
in an interface.



Fig. 8 Use of GUI in computer graphics

1.3 COMPUTER GRAPHICS SOFTWARE
 Graphics software can be classified into two categories:
a. General Programming Packages
b. Special Purpose Applications Packages
a. General Programming Packages
o They are like programming languages; they contain rich set of
graphics functions.
o Basic functions include:
 Generate picture components like circles, polygons, straight lines,
etc.
 Filling colors and setting intensity values
 Selecting views
 Applying transformations
o E.g., graphics library system, Open GL, Open CV
b. Special Purpose Application Packages
o These set of graphics packages are designed for non -programmers.
o Users can create their own graphics without worrying about the
underlying concepts.
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o The interface to the graphics routines in such packages allows users
to communicate with the programs in their own terms.
o E.g., Paint, CAD, etc .

1.3.1 Coordinate Representation :
 General graphics packages are generally designed to be used with
Cartesian coordinate specifications.
 The coordinate values for an object should be converted to Cartesian
coordinates if they were specified in some other reference frame, before
they are passed as input to a graphics package.
 Special purpose packages allow the use of other coordinate systems
based upon the applications used.
 Various Cartesian reference frames can be used to construct and display
a scene.
 Individual shape of an object can be construc ted in a scene within
separate coordinate reference frames called modelling coordinates or
local coordinates or master coordinates.
 Objects can be placed into appropriate positions within a scene using a
reference frame called as world coordinates, once the individual object
shapes have been specified.
 For display, the world coordinate description of the scene is transferred
to one or more output device reference frames, called as device
coordinates or screen coordinates.
 Before final conversion to specific device coordinates, a graphics
system first converts world coordinates positions to normalized device
coordinates in the range from 0 to 1.
 This is done to ensure that the system is independent of any device used
as a workstation.

Fig. 9 Viewing Pipeline

1.3.2 Graphics Functions :
 Users are provided with various functions for creating and
manipulating pictures by general -purpose graphics package.
 Output primitives : These are the basic building blocks for pictures. It
includes string and entities like points, straight lines, curved lines,
circles and polygons.
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 Attributes : The properties of output primitives are referred to as
attributes that describes how a primitive is to be displayed. It includes
line styles, color specifications, intensity of colors and text styles.
 Geometric transformations : It is used to change the size, orientation
or position of an object within a scene.
 Viewing transformations : It is used to s pecify view that is to be
presented and portion of the output display area that is to be used.
 Input functions : It is used to control and process the data flow from
interactive devices.
 Control operations : It is used in clearing a display screen or
initial izing parameters .

1.3.3 Software Standards :
 The primary intension of standardization of graphics software is
portability.
 The packages are designed with standard graphics functions so they can
be used with different hardware systems and used in different
implantation and applications.
 The Graphical Kernel System (GKS) was developed to adopt the first
graphics software standard by the International Standard Organization
(ISO).
 Another graphics standard developed and approved was Progra mmers
Hierarchical Interactive Graphics System (PHIGS ), which is an
extension of GKS.
 It provides improved capabilities for object modelling, color
specifications, surface rendering and picture manipulation.
 An extension of PHIGS, called as PHIGS+, was developed to provide
3D surface shading capabilities.
 PHIGS does not provide a standard methodology for graphics interface
to output devices and does not specify methods for storing and
transmitting pictures.
 Computer Graphics Interface (CGI) : system provides standards for
device interface methods and the Computer Graphics Metafile
(CGM) system specifies standards for archiving and transporting
pictures.

1.4 INPUT DEVICES
1.4.1 Keyboards :
 To enter any kind of string or number, an alphanumeric keyboard is
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 A general keyboard can consist of 84 keys or 101 keys.
 A keyboard consists of numeric keys (0 to 9) , alphabetic keys (A to Z),
control keys (Ctrl, Shift, Alt), special sy mbol keys (!, @, #,”, ?), cursor
control keys (arrow keys), function keys (F1 to F9) .
 Keyboards are not only used to enter any numbers, or symbols or
alphabets, it is also be used for menu selection.
 A numeric keypad is on the right of keyboard for fast entry of numeric
data.

Fig. 10 Keyboard
1.4.2 Mouse :
 To position the cursor on the screen, mouse is used.
 The roller present at the bottom of the mouse records the direction of
the movement .
 The motion of the mouse can also be detected by option sensor in
which the mouse is moved over a special mouse pad that has a grid of
horizontal and vertical lines.
 The optical sensor detects movement across the lines in the grid.
 The mouse consists of two or three buttons on the top to execute any
kind of operation.
 There is a type of mouse called as Z mouse which can be used to pick
an object, rotate that object, move the object in any direction and
navigate our view through a three dimensional scene.


Fig. 11 Mouse


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1.4.3 Trackball a nd Spaceball :
 A trackball is a ball that is used to rotate with palm of the hand or with
the fingers , to produce screen -cursor movement.
 A resistor with sliding or ro tating contact ( called potentiometers)
attached to the ball, is used to measure the amount and direction of
rotation.
 Trackballs are generally mounted on keyboards and is a two -
dimensional positioning device.
 A spaceball does not actually move.
 A sensor whose resis tance varies when force applied ( called strain
gauges ) measure s the amount of pressure applied to the spaceball to
provide input for spatial positioning and orientation as the ball is
pushed or pulled in various directions.
 Spaceballs are used for three -dimensional positioning and selection
operations in CAD, animati on, virtual -reality s ystems, modeling and
other applications.



Fig. 12 Trackball and Spaceball

1.4.4 Joysticks :
 A joystick consists of a small, vertical lever (called the stick) mounted
on a base that is used to move the screen cursor in all directions .
 Position on the screen is selected with actual stick movement or by
applying pressure on stick.
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 Joysticks can be standalone device or can be mounted on a keyboard.
 The direction of the screen -cursor movement depends upon the
direction in which the stick is moved.
 Potentiometers mounted at the base of the joystick measure the amount
of movement, and springs return the stick to the center position when it
is released.

Fig. 13 Joystick

1.4.5 Data Glove :
 Data glove is used to grasp a virtual object.
 The glove is made up of many sensors that helps to detect motions of
fingers and hand .
 The information about the position and orientation of hand is provided
by the e lectromagnetic coupling between transmitting and receiving
antennas .
 Input from the glove can be used to position or manipulate objects in a
virtual scene.
 A two -dimensional projection of the scene can be viewed on a video
monitor .

Fig. 14 Data Glove

1.4.6 Image Scanner :
 Drawings, graphs, color and black -and-white photos, or text can be
stored for computer processing with an image scanner by passing an
optical scanning mechanism over the information to be stored.
 Once we have the internal representation of a picture, we c an apply
transformations to rotate, scale, or crop the picture to a particular screen
area.
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 For scanned text input, various editing operations can be performed on
the stored documents.
 Some scanners are able to scan either graphical representations or text,
and they come in a variety of sizes and capabilities.


Fig. 15 Image Scanner

1.4.7 Touch Panels :
 A type of display screen that has a touch -sensitive transp arent panel
covering the screen are called touch panels.
 The input is recorded through finger when in contact with the screen.
 It allows a user to interact directly what is being displayed instead of
using mouse and keyboards.
 Generally used on smart phones and laptops.


Fig. 16 Touch Panel

1.4.8 Light Pens :
 It is a pencil -shaped device used to select screen positions by detecting
the light coming from points on the CRT screen.
 Also used to select menu driven applications or draw pictures on
monitor screen.
 It consist of photocell and optical system placed in a small tube.
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 When its tip is moved over the monitor screen, the photocell -sensing
element detects screen location and sends corresponding signals to
CPU.


Fig. 17 Light Pens

1.4.9 Voice Systems :
 It is used in some graphics workstations as input devices to accept
voice commands.
 These systems operate by matching an input against a predefined
dictionary of words and phrases.
 It is easy to use and most efficient device.
 E.g., a microphone.


Fig. 18 Voice Systems

1.5 ACTIVE AND PASSIVE GRAPHICS Active Graphics Passive Graphics Also known as interactive graphics. Also known as non-interactive graphics. In active graphics, users have control over the picture displayed on the screen. In passive graphics, users do not have any control over the picture displayed on the screen. These graphics include animations. These graphics do not change its shape on display. It is a two-way communication between the computer and the user. It is a one-way communication between the computer and the user.
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Low cost. High cost. Image quality produced is high. Image quality produced is not too good. E.g., video games, drawing on touch screens E.g., titles on TV screen, data representation through charts
1.6 SUMMARY
 Computer graphics is a way in which pictures or object can be created
and manipulated.
 It can be used in creation of 2D images, 3D images or animations.
 The applications of computer graphics can be in the field of education,
training, CAD systems, entertainment, presentations, computer art,
image processing and many more.
 The use of various graphics functions with graphics standard make up a
graphics software.
 Various devices like keyboard, mouse, data glove, joysticks, etc. can be
used to provide input to any graphics system.

1.9 UNIT END EXERCISE
1. Define computer graphics.
2. List and explain any five applications of computer graphics.
3. Write a short note on computer graphics software.
4. Explain various graphics software standards.
5. Explain various graphics functions.
6. List and explain any five input devices used in graphics.

1.7 REFERENCES
 D. Hearn and M. P. Baker, Computer Graphics , 2nd Edition, 1997.

1.8 BIBLIOGRAPHY
 J. Hughes, A. Van Dam, M. MCGuire, D. Sklar, J. Foley, and K.
Akeley, Computer Graphics, Principles and Practice , 3rd Edition,
Pearson, 2014.
 P. Shirley and S. Marschner, “Fundamentals of Computer Graphics:
Ray Tracing,” CRC Press . p. 752, 2009.


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2
DISPLAY TECHNOLOGIES

Unit Structure
2.0 Objectives
2.1 Display Devices
2.1.1 Storage Tubes Graphics Displays
2.1.2 Calligraphic Refresh Graphics Displays
2.1.3 Raster Refresh Graphics Displays
2.1.4 Random Scan Display v/s Raster Scan Display
2.1.5 Cathode Ray Tube (CRT)
2.1.6 Color CRT
2.1.6.1 Beam -Penetration Technique
2.1.6.2 Shadow Mask Technique
2.1.7 Liquid Crystal Display (LCD)
2.1.8 LCD Displays v/s CRT Monitors
2.2 Raster Scan Systems
2.2.1 Video Controller
2.2.2 Display Processor
2.3 Random Scan Systems
2.4 Summary
2.5 Unit End Exercise
2.6 References
2.7 Bibliography

2.0 OBJECTIVES
After this chapter, student will be able to:
 Identify various display devices .
 State the working of display devices.
 State and differentiate raster scan and random scan devices.

2.1 DISPLAY DEVICES
 There are number of display devices used in graphics.
 Output devices is another name of display devices.
 The display devices generally used are CRT and LCD.

2.1.1 Storage Tubes Graphics Displays:
 This is also known as Direct View Storage Tube s (DVST). munotes.in

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 The raster scan display refreshes the screen to maintain a screen image
whereas DVST give s alternate method of maintaining the screen image
without refreshing.
Components of DVST :
i) Two Electron Guns
ii) Phosphor coated screen,
iii) Storage grid mesh
iv) Collector

Fig 1. Working of Direct View Storage Tube

Working Principle:
1. There are two electron guns, namely primary gun and flood gun.
2. The primary gun stores the picture pattern and the flood gun
maintains the picture display.
3. A primary gun produces high-speed electrons, which strike on the
storage grid to draw the picture pattern.
4. Continuous low speed electrons from flood gun pass t hrough the
control grid and got attracted to the positive charged areas of the
storage g rid.
5. The low speed electron then penetrate s the storage grid and strike
the phosphor coating.
6. During this process, the collector behind the storage grid smooths
out the flows of food electrons.

Merits:
 It has flat screen.
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 Refreshing of CRT is not required.
 Complex picture can be displayed on high resolution.

Demerits:
 The performance of DVST is poor compared to CRT.
 Erasing requires removal of charge on the storage grid.
 Selective or part erasing of screen is not possible.
 Erasing of screen produces flash over the entire screen.


2.1.2 Calligraphic Refresh Graphics Displays:
Calligraphic refresh graphic displays are also referred as random scan or
stroke -writing or vector displays.
Components:
A CRT display unit with electron beam,, Refresh display files,
Graphics controller
Working Principle:
 A CRT, as a random scan display unit, has an electron beam directed
only to the parts of the screen where a picture is to be drawn.
 Random scan monitors draw a picture one line at a time.
 The c omponent of a picture can be drawn and refreshed by a random
scan system in a specified order.
 A picture definition is stored as a set of line -drawing commands in an
area of memory called a refresh display file or refresh buffer.
 To display a specified pic ture, the system cycles through the set of
commands in the display file, drawing each component line one by one.
 After all line drawing commands have been processed, the system
cycles back to the first line command in the list and repeat the
procedure of s can, display, and retrace.
Example :
 A pen plotter


Fig 2. Working of Random Scan Systems
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 Random scan displays draw all the component lines of a picture with 30
to 60 times for each second.
 High quality vector systems are capable of handling approximately
100,000 short lines at this refresh rate.
 The faster refreshing of the set of lines could burn out the phosphor.
Therefore, when a small set of lines are to be displayed, each refresh
cycle is delayed to avoid greater refresh rates, typically 60 frames per
second.
 Random scan systems are designed for line drawing applications; hence
it cannot display realistic shaded scenes.
 Vector displays have higher resolution than raster systems, as picture
definition is stored as a set of line drawing instructions instead of a set
of intensity values for all screen points.
 These vector displays produce smooth line drawings, because the CRT
beam directly follows the line path.

2.1.3 Raster Refresh Graphics Displays :
A raster scan display is the most common m ethod of drawing
images on a CRT screen.
Working Principle:
 In this method, horizontal and vertical deflection signals are generated
to move a beam all over the screen in a pattern for displaying any
image.
 The electron beam is swept across the screen one row at a time from
top to bottom.
 The electron beam sweeps back and forth from left to right across the
screen.
Horizontal Retrace
While electron beam moves from left to right the beam is on and when it
moves back from right to left the beam is off. This p henomenon is known
as horizontal retrace.
Vertical Retrace
As soon as the beam reaches the bottom of the screen, it is turned off and
is rapidly retraced back to the top to start again. This is known as vertical
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Fig 3. Horizontal and vertical retra

 Raster scan displays maintain the steady image on the screen by
repeating scanning of the scan image. This process is known as
refreshing the screen.


Fig 4. Working of Raster Scan Systems


2.1.4 Random Scan Display V/S Raster Scan Display:
Random Scan Display Raster Scan Display It has high resolution. It has less resolution. The electron beam is directed on
portion of screen where picture
is to be displayed. The electron beam is directed to
the whole screen i.e. scanning
one row at a time. It is expensive . It is less expensive. It stores picture definition as a
set of line commands called
refresh display file. It stores picture definition in
refresh buffer or frame buffer. Smooth line is produced. Zigzag line is produced. Hard to display realist ic image . Realistic image is displayed. munotes.in

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2.1.5 Cathode Ray Tube (CRT ) :
 A CRT is an evacuated glass tube.
 An electronic gun at the rear of the tube produces a beam of electron,
which is directed towards the front of the screen by a high voltage.
 The inner side of the screen is coated with phosphor substance, which
gives off light when it is stroked by electron.
 The control grid voltage determines the velocity achieved by the
electrons before they hit the phosphor.
 The control grid voltage determines how many electrons are actually in
the el ectron beam.
 Thus, control grid controls the intensity of the spot where the electron
beam strikes the screen.
 The focusing system concentrates the electron beam so that the beam
converges to a small point when it hits the phosphor.
 The deflection system of CRT consists of two pairs of parallel plates,
namely horizontal and vertical deflection plates.
 The voltage applied to vertical deflection of the electron beam and the
voltage applied to the horizontal deflection plates controls the
horizontal deflecti on of the electron beam.
 There are two techniques used for producing images on CRT screen,
namely Random Scan and Raster Scan.

Fig 5. Working of CRT Monitors

It uses line drawing applications. It uses pixels along scan line to
draw an image. Refresh rate is 30 to 60
frames/second. Refresh rate is 60 to 80
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2.1.6 Color CRT :
 A CRT monitor displays color pictures by using a combination of
phosphors that emit different -colored light.
 It generates a range of colors by combining the emitted light from
different phosphors.
 There are two basic techniques used for producing color displays:
o Beam -penetration technique
o Shadow mask technique

2.1.6.1 Beam -Penetration Techniques :
 This technique is used with random -scan monitors.
 In this technique, the inside of CRT screen is coated with two layers of
phosphor, usually red and green.
 The displayed color depends on how far the electron beam penetrates
into the phosphor la yers.
 The outer layer is of red phosphor and the inner layer is of green
phosphor.
 A beam of slow electrons excites only the outer red color.
 A beam of very fast electrons penetrates through the red layer and
excites the inner green layer.
 At intermediate beam speeds, combination of red and green light are
emitted and two additional colors, orange and yellow are displayed.
 The beam acceleration voltage controls the speed of electrons and
hence the screen color at any point on the screen.

Merit s:
 It is inexpensive technique to produce color in random scan monitors.

Demerit s:
 It can display only four colors.
 The quality of picture produced by this technique is not good as
compared to other techniques.

Fig 6. Beam Penetration Technique
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2.1.6.2 Shadow Mask Technique:
 The shadow mask technique produces a much wider range of colors
then the beam penetration technique.
 Hence, this technique is commonly used in raster -scan displays
including color TV.
 In a shadow mask technique, CRT has three phospho r color dots at
each pixel position.
 One phosphor dot emits a red light, second emits green light and third
emits blue light.
 It has three electron guns, one for each color dot and a shadow mask
grid just behind the phosphor -coated screen.
 The shadow mask grid consists of series of holes aligned with the
phosphor dot pattern.
 Three electron beams are deflected and focused as a group on to the
shadow mask and when they pass through a hole in the shadow mask,
they excite a dot triangle.
 A dot triangle consis ts of three small phosphor dots of red, green and
blue color.
 A dot triangle when activated appears as a small dot on the screen,
which has color of combination of three small dots.
 By varying the intensity of three electron beams, we can obtain
different colors in the shadow mask CRT.

Fig 7. Shadow Mask Technique

2.1.7 Liquid Crystal Displays (Lcd):
 The term liquid crystal refers to the fact that these compounds have a
crystalline arrangement of molecules, yet they flow like a liquid.
 Two glass plates, each containing a light polarizer at the right angles to
the other plate squeezes the liquid -crystal material.
 Rows of the horizontal transparent conductors are built into one glass
plate and columns of vertical conductors are put into the other plate.
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 The inte rsection of two conductors defines a pixel position.
 In the ON state, polarized light passing through material is twisted so
that it will pass through the opposite polarizer.
 It is then reflected back to the viewer.
 To turn OFF the pixel, we apply a voltag e to the two intersecting
conductors to align the molecules so that light is not twisted.

Merits:
 Less bulky.
 Suitable component for portable devices like laptops, mobile phones,
etc.
 Power consumption is much less so ideal to be operated by batteries.

Demerits:
 More expensive than the CRT and more difficult to manufacture.


Fig 8. Working of LCD

2.1.8 Lcd Displays V/S Crt Monitors:
LCD Displays CRT Monitors Thin, compact, lightweight.
Takes up less space. Bulky, heavier and takes up
more space. Lack in brightness than CRT. Appreciable amount of
brightness. Consumes less electricity and
produce less heat. Consumes more power and
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2.2 RASTER SCAN SYSTEMS  Hardware architecture of the raster sc an system is shown in the Fig 9 .
 In addition to CPU, a special -purpose processor, called video controller
or display controller, is used to control the operation of the display
device.
 In this architecture, frame buffer is allocated a dedicated area in system
memory.
 Frame buffer can be anywhere in the system memory.


Fig 9. Architecture of a Raster Scan System

 Video controller reads the command from frame buffer and draw pixel
accordingly.

2.2.1 Video Controller:
 Another variant of a raster scan system is shown in F ig 10 .
 A dedicated memory area is allocated to the frame buffer.
 Video controller is given direct access to frame buffer.
 The organization can render scene quickly because video controller do
not have to put a request on system bus to read intensity value from
frame buffer.
 In some system, origin is set at the bottom left location of the screen.
 X coo rdinate increases from left to right and Y coordinate increases
from bottom to top.

Much expensive. Less expensive. Portable. Not portable. Restricted viewing angles. I t
needs to be viewed from front to
have better view. Viewable from almost every
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Fig 10. Architecture of a Raster Scan System

 In some systems, origin is set at the top left corner. In such case, Y
coordinate increases from top to bottom.
 Two register are maintained to store the coordinates of pixel on screen.
 Intensity value from frame buffer is retrieved from current (x, y)
location.
 After painting each pixel, x register is incremented by 1 and process is
repeated until x value hit to x max.
 After processing all the pixels on current scan line, content of x register
is set to 0, and y is incremented by 1, and video controller processes the
same way.
 Refresh rate is 60 frames/second.
 This is shown in F ig 11 .
 In high quality systems, use two fr ame buffers, so that one buffer can
be used for refreshing while other is being filled with intensity values.
 Then both buffer switch the role.
 This mechanism is used in real time animation because it does not
waste time in reloading the buffer.
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Fig 11. Refresh operation on video controller

2.2.2 Display Processor :
 Display processor is also known as graphics controller or a display
coprocessor.
 Its purpose is to free the CPU from graphics operations.
 For this, a separate display processor memory area is provided as
shown in F ig 12 .
 The major task of a display processor is to digitize a picture definition
given in an application program into a set of pixel intensity values for
storage in frame buffer.
 This digitize process is called scan -conversion.
 Graphi cs commands specify straight lines and other geometric objects
are scan converted into a set of discrete intensity points.
 Scan converting a straight line means, we have to locate the pixel
positions closest to the line path and store the intensity for each position
in the frame buffer.
 This method can also be used to scan convert curved lines or polygon
outlines.
 Display processor can also be used to generate various line styles,
displaying color areas, performing transformations and manipulating
object s.
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Fig 12. Raster Scan System with a display processor

2.3 RANDOM SCAN SYSTEMS
 Hardware architecture of random scan system is shown in F ig 13 .

Fig 13. Random Scan System

 Application program resides in system memory.
 Graphics packages translate the graphics commands in the application
program into a display file.
 Display file is also stored in system memory.
 Display processor accesses the content of display file and display on
monitor screen.
 Display processor retrieves one by one command from display file and
draw it on the screen.
 Display processor in random scan system is also called a graphics
controller or display processing unit.
 Random scan devices render the scene using short line.
 Electron beam track the line position directly, rather t han going through
each and every pixel on screen.
 Entire scene is defined using segment of line in in application program.
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 Line segments are defined by pair of end points.
 A scene is then drawn one line at a time by positioning the beam to fill
the line b etween specified points.

2.4 SUMMARY
 The display devices are the means of output devices.
 Random scan displays stores picture definition as a set of line
commands .
 Raster scan displays plots image pixel by pixel on the screen.
 The main components of CRT are electron gun, control grid, focusing
system, horizontal and vertical deflection plates.
 Color CRT uses two techniques: beam penetration and shadow mask.
 In beam penetration technique, tw o colors red and green are used which
generates orange and yellow. So only four colors are produced.
 In shadow mask technique, the three -phosphor dots pr oduced are red,
green and blue, which produces more number of colors.
 LCD are portable that consumes less electricity.

2.5 UNIT END EXERCISE
1. Differentiate between raster scan and random scan displays.
2. Explain the working of CRT.
3. Explain the shadow mask technique.
4. Explain the raster scan system.
5. Explain random scan systems.
6. Differentiate between LCD and CRT.

2.6 REFERENCES
 D. Hearn and M. P. Baker, Computer Graphics , 2nd Edition, 1997.

2.7 BIBLIOGRAPHY
 J. Hughes, A. Van Dam, M. MCGuire, D. Sklar, J. Foley, and K.
Akeley, Computer Graphics, Principles and Practice , 3rd Edition,
Pearson, 2014.
 P. Shirley and S. Marschner, “Fundamentals of Computer Graphics:
Ray Tracing,” CRC Press . p. 752, 2009.
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3

SCAN CONVERSION –
LINE AND CIRCLE

Unit Structure
3.0 Objectives
3.1 Scan Conversion of Points
3.2 Scan Conversion of Lines
3.2.1 DDA Line Drawing Algorithm
3.2.2 Example of DDA Line Drawing Algorithm
3.2.3 Bresenham’s Line Drawing Algorithm
3.2.4 Example of Bresenham’s Line Drawing Algorithm
3.3 Scan Conversion of Circle
3.3.1 Midpoint Circle Algorithm
3.3.2 Bresenham’s Circle Algorithm
3.3.3 Midpoint Ellipse Algorithm
3.4 Aliasing and Problems in aliasing
3.5 Summary
3.6 Unit End Exercise
3.7 References
3.8 Bibliography

3.0 OBJECTIVES After studying this chapter, student will be able to:
 State and explain various line drawing algorithms
 Generate lines based on line algorithms
 State and explain circle and ellipse algorithms
 Generate circle based on algorithms
 Define aliasing and its problems

3.1 INTRODUCTION OF SCAN CONVERSION
Scan conversion :
Scan conversion is defined as a process of representing continuous
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 The graphics objects can be a point, line, circle, arc, re ctangle, polygon
and other shapes .
 Graphics objects are continuous.
 Pixels are always discrete. Each pixel has t wo states: on and off. 0 is
addressed by pixel off and 1 is addressed by pixel on.
 The video display device of the computer has a circuit that converts the
binary values of 0 and 1 into the information of pixel on and off.
 Image is represented as collection of pixels.

Pixel :
 Pixel is a smallest unit of a digital image or graphics object, which can
be displayed and represented on a display device. A pixel is also called
as a pel or picture element.
 A pixel is a bas ic logical unit in the graphics and the number of pixel
are combined to form a complete image, video or text.
 A pixel is represented by a dot or a square on a display screen.
 Pixels are the basic building blocks of an image that are generated
using geometric coordinates.
 Depending on the display monitor and graphic s card, the size and color
combination of pixel varies and is measured in terms of display
resolution.

Fig 1. Pixel representation

3.2 SCAN CONVERSION OF POINTS
 Vectors are defined mathematically in an infinite, real number
Cartesian Co -ordinate system.
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 The physical coordinates of the pixel on the computer screen based
upon the screen resolution is called as screen coordinates, device
coordinates, or pixel coordinates.
 A display device deals with finite, discrete coordinates.
 Both x -coordinate and y -coord inate are positive values.
 On a display device, the origin (0, 0) is always on the top -left corner of
the screen.

Fig 2. A pixel representation as screen coordinates on a display
device
 The values in the x -coordinate increases from left to right and the
values of the y -coordinate increases from top to bottom.
 A mathematical point (x, y) where x and y are real numbers within an
image area, needs to be scan -converted to a pixel location at (x’, y’).
 Each pixel on graphic display does not repre sent a mathematical point
like P (2.9, 4.22).
 This may be done by making x’ and y’ to be the integer part of x and y,
by applying any mathematical function.
Example: ,
Floor is a function that returns largest integer that is less than or equal
to the argument
x’=Floor(x)
y’=Floor(y)
for point P (2.9, 4.22).
x’=Floor (2.9) and y’=Floor (4.22),
(x’, y’) will be (3, 4).

3.3 SCAN CONVERSION OF LINES
A line has two end points and is defined by an equation y=mx + c, where
m is the slope of the line and c is the y intercept .
Consider the two end points as P1 (x1, y1) and P2 (x2, y2). The equation
of the line is used to determine the x and y coordinates of all the points
that lie between P1 and P2 as shown in Fig 3 .
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The line is drawn on the screen when there are two end points as P1
and P2, and the computer fills it with pixels by the i ntermediate point
values. This can be done with various line drawing algorithm s like
DDA and Bresenham’s .

Fig 3 . Line representation on coordinate system

3.3.1 DDA Line Drawing Algorithm
DDA stands for Digital Differential Analyzer. This algorithm works on
incremental approach.
Drawing Concept
 The next coordinate to be plotted is calculated based on the previous
coordinates.
 The line to be drawn is based on the analysis of difference between the
pixel points.
 The incremental points are calculated based upon the start point and the
end point of the line.
 It calculates pixel positions along a line by taking unit ste p increment
along one direction and calculating corresponding coordinate position
based on the rate of change of the coordinate ( ∆x or ∆y).
 The slop e of the line is the ratio of difference of y coordinates and
difference of x coordinates.
∆y = (y2 – y1)
∆x = (x2 – x1)
where,
(x1, y1) is the start point
(x2, y2) is the end point
∆y is the difference between y coordinates
∆x is the difference between x coordinates
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 The slope is calculated as:

 The DDA algorithm is based on the values of ∆x and ∆y.
 The value of the slope will be either positive or negative.
 If the value of the slope is positive, then ∆x and ∆y are incremented,
else decremented.
 For each pixel we move right (along the x-axis), we need to move
down (along the y -axis) by m pixels (based on screen coordinates) .

Algorithm:
1. Start
2. Input the two end points of a line as (x1, y1) and (x2, y2)
3. Compute
dx=x2 -x1 and dy=y2 -y1
4. if abs(dx) > abs(dy)
step = abs(dx)
else
step = abs(dy)
5. Compute
xinc = dx/step
yinc = dy/step
6. Initialize x = x1 and y = y1
7. Plot pixel (x, y)
8. For k=1 to step
{
x = x + xinc
y = y + yinc
Plot pixel (ROUND(x), ROUND(y));
}
9. End

Merits:
 It is simple.
 It is fast to implement.

Demerits:
 It involves floating values, which requires rounding off to plot on
screen.
 Rounding to integers takes time.
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3.3.2 Example o f DDA Line Drawing Algorithm:
 Consider a line PQ with P = (0, 0 ) and Q = (7, 4). Apply the simple
DDA algorithm and calculate the pixel on this line.
Solution to trace line PQ .
1. Start
2. Input x1=0, y1=0 , x2=7, y=4
3. Compute
dx=x2 -x1=7 -0=7
dy=y2 -y1=4 -0=4
4. abs(dx) =7 and abs(dy)=4, so abs(x)>abs(y)
so, step=abs(dx) =7
5. Compute
xinc=dx/step=7/7 =1
yinc=dy/step=4/7=0.6
6. Initialize x=0, y=0
7. Plot the pixel ( 0, 0)
8. For k=1 to 7 ,
Calculate new values of x and y as x = x + xinc and y = y + yinc.
Calculated values shown in Table 1.
9. End
Table 1. Pixel value to compute line PQ
k Plot x y 0, 0 0 0 1 1, 1 1 0.6 2 2, 1 2 1.2 3 3, 2 3 1.8 4 4, 2 4 2.4 5 5, 3 5 3 6 6, 4 6 3.6 7 7,4 7 4.2
 Fig 4 shows the screen plot of line PQ. munotes.in

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Fig 4. Screen plots of line PQ
3.3.3 Bresenham’s Line Drawing Algorithm:
Bresenham’s line drawing algorithm determines the points of an n-
dimensional raster that should be selected in order to form a close
approximation to a straight line between two points.
This algorithm provides the fast and efficient way t o represent continuous
line on a discrete plane of a computer display.
The algori thm is suitable for digital plotter and CRT.
Algorithm:
1. Start
2. Input the line end points (x1, y1) and (x2, y2)
3. Plot pixel (x1, y1)
4. Compute
dx = x2 – x1
dy = y2 – y1
5. Initialize p k = 2*dy – dx
6. For each x along the line, starting at k = 0, perform the following
test
If p k< 0,
Plot pixel (x k+1, y k)
pk+1= p k + 2*dy
Else
Plot pixel (x k+1, y k+1)
pk+1= p k + 2*dy –2*dx
7. Repeat step 6, dx – 1 times.
8. Plot (x2, y2)
9. End

Merits:
 It generates only integer values, so simple to use.
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 It is faster than DDA line drawing algorithm.
 It avoids generation of duplicate points.
 Points generated are more accurate.

Demerits:
 The resultant line obtained is not smooth.
 This algorithm is for basic line drawing. It cannot handle zigzag lines.
3.3.4 Example of Bresenham’s Line Drawing Algorithm:
 Consider a line AB with A = (0, 0) and B = (7, 4). Apply the
Bresenham’s line algorithm and calculate the pixel on this line.
 Solution to trace line AB.
1. Start
2. Input x1=0, y1=0, x2=7, y=4
3. Plot (0, 0)
4. Compute
dx=x2 -x1=7 -0=7
dy=y2 -y1=4 -0=4
5. Initialize pk = 2*dy -dx,
pk = 2*4 -7=8-7=1
6. For k=0 to 6,
Calculate new values of x and y based upon decision parameter
pk
If p k< 0,
increment x, y will be same, plot (xk+1, y k)
compute p k+1 =p k + 2*dy
else
increment x and y, plot (xk+1, y k+1)
computep k + 1 = pk+ 2*dy – 2*dx
Calculated values shown in Table 2 .
7. Repeat the steps dx -1=7-1=6 times
8. Plot (7, 4)
9. End

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1 -5 3 2, 1 2 1.2 2 3 -3 3, 2 3 1.8 3 -3 5 4, 2 4 2.4 4 5 -1 5, 3 5 3 5 -1 7 6, 4 6 3.6 6 7 1 7,4 7 4.2  Fig 5 shows the screen plot of line AB .

Fig 5 . Screen plots of line AB

3.4 SCAN CONVERSION OF CIRCLE
A circle is defined as a set of points that are at the same distance from
center and the distance from center to any point on the circumference of
the circle is called a radius.
 It is a n eight -way symmetry as in Fig 6 , which can be divided into
four quadrants, and each quadrant into two octants.
 This symmetry helps in implementation of any circle -drawing
algorithm.

Fig 6.Midpoint criteria
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 The symmetric property of the circle is utilized to scan convert the
circle by plotting eight points for eight octants.
 This means for every point, eight poin ts can be plotted as (x, y), (y,
x), (-y, x), ( -x, y), ( -x, -y), (-y, -x), (y, -x) and (x, -y) as shown in Fig
6.
 The entire circle can be obtained by just computing the first octant
and then repeat the reflection of the point about each 45o axis.
 It produces seven more points on the circumference of the circle.
Hence, it is sufficient to calculate only one octant.
 The circle centered at the origin can be described by the equation:
x2 +y2 = r2
where,
x and y are the coordinates
r is the radius
 The second approach describe the circle using trigonometric relation,
where x and y coordinates of the circle is given by x=r cosθ and y =r
sinθ, and θ is the current angle.

Fig 7 . Representing circle in trigonometric relations

3.4.1 Midpoint Circle Algorithm:
This algorithm is used to determine the points needed for rasterizing
the circle.
 The perimeter points will be calculated for first octant and will be
printed along the mirror points in other octants.
 For any given point (x, y), the next point to be plott ed can be (x, y+1)
or (x -1, y-1) based upon the three conditions.
 Given a circle centered at (0, 0), with radius r and point P(x, y), f(P)=x2
+y2 - r2
o If f(P) < 0, the given point lies inside the circle boundary, then the
upper pixel can be chosen.
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o If f(P ) = 0, the given point lies on the circle boundary, then any
pixel can be chosen.
o If f(P) > 0, the given point lies outside the circle boundary, then
the lower pixel is chosen.

Algorithm:
1. Start
2. Declare x, y, r, P, x c, yc where (x c, yc) are center coordinates, x and y
points to be plotted, r is the radius, P is decision factor.
3. Initialize x = 0 and y = r.
4. Compute decision factor P = (5/4) – r.
5. Repeat the steps while x ≤ y
6. Plot (x, y)
7. If P < 0
Set P = P + 2x +3
Else if P ≥ 0
Set P = P + 2(x -y) + 5
y = y – 1
8. do x = x + 1
9. End

Merits:
 It is an efficient algorithm.
 It is easy to implement.
 It is based on simple circle equation x2 + y2 = r2
 It helps to create curves on raster display.

Demerits:
 It is a time consuming algorithm.
 Sometimes the point of the circle are not accurate.

3.4.2 Bresenham’s Circle Algorithm :
 The Bresenham’ s circle drawing algorithm calculates all the points
nearest to the circle boundary.
 It is an incremental method where one coordinate of the point is
incremented and another coordinate is calculated according to it.
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 The strategy is to select the pixel, which is at the least distance from
the true circle boundary, and then keep calculating the succes sive
points on the circle.
 The circle follows the symmetric property.
 In this algorithm, at any point (x, y) we have two options either to
choose the next pixel at the top i.e. (x+1, y) or at the bottom i.e. (x+1,
y-1).
 This is done using decision paramete r d where,
If d > 0, (x+1, y -1) will be the next pixel to plot
Else (x+1, y) will be the next pixel
 To draw a circle with center (x c, yc) and radius r, it will start from (0, r)
and move in first quadrant till x=y with in itial conditions:
d=3-2*r
x=0 and y=r

 Algorithm:
1. Start
2. Declare x, y, r, D , xc, yc where (x c, yc) are center coordinates, x and
y points to be plotted, r is the radius, D is decision factor.
3. Calculate decision parameter D as : D = 3–(2 * r)
4. Initialize x = 0, y = r
5. Compute next pixels on circle based upon decision parameter
While x ≤ y
Plot (x, y)
if D<0 , then
D = D + 4x + 6
else
D = D + 4(x - y) + 10
y = y – 1
end if
x = x + 1
6. End

Merits:
 It is a simple algorithm.
 It is easy to implement.
 It is based on simple circle equation x2 + y2 = r2
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Demerits:
 It is not suitable for high graphics and complex images.
 Accuracy issues while generating points.

3.4.3 Midpoint Ellipse Algorithm:
Ellipse is defined as the geometric figure, which is set of all points
on a plane whose distance from two fixed points known as foci remains
constant.
 It has two axis, namely major axis and minor axis, where the major
axis is the longest diameter and minor axis is the shortest diameter.
 Ellipse has a four -way symmetry property, which means quadrants are
symmetric an d not the octants.
 Midpoint ellipse algorithm plots the point of an ellipse on the first
quadrant by dividing the quadrant into two regions.
 Each point (x, y) is projected to other three quadrants ( -x, y), (x, -y)
and ( -x, -y) as it uses four -way symmetry .


Fig 8 . Four -way symmetry and Criteria for Ellipse

 Consider the equation of ellipse as f(x, y) = r y2x2 + rx2y2 – rx2ry2
where,
rxis the x radius
ryis the y radius
 For any point (x, y) the next point to be plotted is based on the three
conditions as:
o If f(x, y) < 0, then (x, y) is inside the ellipse
o If f(x, y) > 0, then (x, y) is outside the ellipse
o If f(x, y) = 0, then (x, y) is on the ellipse
 There will be two decision parameters based upon the two regions, P1
for region 1 and P2 for region 2.
 For region 1, P1 = ry2 + rx2 / 4 – ryrx2
 For region 2, P2 = r y2(x + 1 / 2)2 + rx2(y – 1)2 – rx2ry2
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 Algorithm:
1. Start
2. Declare r x , ry , x , y, dx , dy , P1 , P2(r x is x radius, r y = y radius, x =
x coordinate to plotted , y = y coordinate to be plotted, dx , dy , P1=
decision parameter for region 1, P2= decision parameter for region
2)
3. Initialize initial point for region 1 as x = 0, y = r y
4. Calculate P1 = r y2 + rx2 / 4 – ryrx2, dx = 2 r y2x, dy = 2 r x2y
5. Update values of dx and dy after every iterat ion.
6. Repeat while (dx Plot (x, y)
If (P < 0)
x = x + 1
P1 = P1 + r y2 (2*x + 3)
Else
x = x + 1
y = y – 1
7. When dx ≥ dy, plot region 2.
8. Calculate P2 = r y2(x + 1 / 2)2 + rx2(y – 1)2 – rx2ry2
9. Repeat till (y > 0)
If (P2 > 0)
y = y – 1
P2 = P2 – 2*y*r x2 + rx2
Else
x = x + 1
y = y – 1
P2 = P2 + 2*r y2*(2*x) – 2*y*r x2 + rx2
10. End

Merits:
 It is easy to implement.

Demerits:
 It is a time consuming algorithm.



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3.5 ALIASING AND PROBLEMS IN ALIASING
 In computer graphics, smooth curves and other lines become s jagged
because the resolution of the graphics device or file is not high enough
to represent smooth curve.
 In line drawing algorithms, not all rasterized locations match with the
true line and we have to select the optimum raster locations t o
represent a straight line. This problem is more in low -resolution
screens.
 In such screens, lines appear like a stair -step. This effect is called
aliasing.
 The aliasing effect can be reduced by adjusting the intensities of the
pixel along the line.
 This process of adjusting intensities of the pixels along the line to
minimize the effect of aliasing is called antialiasing.
 The aliasing effect can be minimized by increasing resolution of the
raster display device.

3.6 SUMMARY
 Scan conversion is essential method for plotting discrete pixels.
 DDA line drawing algorithm uses floating -point values, which is
converted to integer while plotting pixel. Therefore, it is time
consuming.
 Bresenham’s line drawing algorithm is more efficient and faster than
DDA, as it involves only integer values.
 Midpoint circle algorithm is efficient and easy to implement and works
best for raster displays.
 Bresenham’s circle algorithm is simple to implem ent but do not work
on complex images.
 Midpoint ellipse algorithm works on criteria of two regions and is time
consuming.
 Aliasing occurs on low resolution screens, wher e smooth curves or
lines become jagged.

3.7 UNIT END EXERCISE
1. Explain the DDA line drawing algorithm.
2. Explain the Bresenham’s line drawing algorithm with example.
3. Given the endpoints of line XY as (2, 3) and (8, 9). Apply DDA line
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4. Explain the midpoint circle algorithm.
5. Write the Bresenham’s circle algorithm.
6. Write the algorithm for midpoint ellipse.
7. Write a short note on aliasing.

3.8 REFERENCES
 D. Hearn and M. P. Baker, Computer Graphics , 2nd Edition, 1997.
 R. K. Maurya, Computer Graphics , John Wiley.

3.9 BIBLIOGRAPHY
 J. Hughes, A. Van Dam, M. MCGuire, D. Sklar, J. Foley, and K.
Akeley, Computer Graphics, Principles and Practice , 3rd Edition,
Pearson, 2014.
 P. Shirley and S. Marschner, “Fundamentals of Computer Graphics:
Ray Tracing,” CRC Press . p. 752, 2009.


*****

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4

SCAN CONVERSION – LINE AND
POLYGON CLIPPING

Unit Structure
4.0 Objectives
4.1 What is clipping?
4.2 Point Clipping
4.3 Line Clipping
4.3.1 Cyrus -Beck Line Clipping Algorithm
4.3.2 Cohen -Sutherland Line Clipping Algorithm
4.3.3 Liang -Barsky Line Clipping Algorithm
4.4 Polygon Clipping
4.4.1 Sutherland -Hodgeman Polygon Clipping
4.5 Summary
4.6 Unit End Exercise
4.7 References
4.8 Bibliography

4.0 OBJECTIVES
After studying this chapter, student will be able to:
 Identify clipping region with region codes
 Implement various clipping algorithms on lines and polygon
 Extract scene with different clipping methods

4.1 WHAT IS CLIPPING?
A procedure that identifies the portions of a picture that are either lying
inside or outside of a specified region of space is referred to as clipp ing
and can be done by using the clipping algorithm s.
 The region against which an object is to be clipped is called a clip
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Fig 1. Clipping window
 Applications of clipping includes extracting part of a defined scene for
viewing, identifying visible surface in three -dimensional views, object
boundary, displaying a multi -window environment and many more.
The clip window can be a general polygon or any curved boundary.
 For viewing trans formation, we want to display only those picture parts
that are within the window area.
 Everything outside the window is discarded.
 Clipping algorithm can be applied in world coordinates, so that only the
contents of the window interior are mapped to the d evice coordinates.
 Clipping can be done on a point, lines, polygons and text.
 In Fig 2 , lines and points within window should be kept (line (P5 -P6)
and point P8) and the line lying outside the window should be clipped
(line (P1 to P2), line (P3 to P4) and point P7) .


Fig 2. Example of clip ping
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4.2 POINT CLIPPING
Point clipping is a process to define the position of the point.
Point clip
Consider the clipping window is a rectangle in t he standard position, a
point P (x, y) is used for display with the following constraints
xwmin ≤ x ≤ xw max AND yw min ≤ y ≤ yw max
where edges of the clip window are xw min, xw max, yw min and yw max then the
point is clipped.
 In Fig 3, point B and point C are within the window range (xw min,
xwmax, yw min, yw max) so they are not clipped, while point A, point D
and point E lie outside the window so they are clipped.


Fig 3. Example of Point Clipping

4.3 LINE CLIPPING
 Removing the portion of the lines which are lying outside the edges of
the edges is called as line clipping .
Procedure for Line Clipping .
1. Test a given line segment to determine whether it lies completely
inside the clipping window,
2. If it does not, we try to determine whether it lies completely outside
the window.
3. If the line segment is not completely inside or completely outside,
perform inters ection calculations with one or more clipping
boundaries , with the use of various line clipping algorithms.

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Fig 4 . Example of Line Clipping
Examples
 In Fig 4, line P1 to P2 lies completely inside the window, so it does not
requires clipping and is saved.
 Line P3 to P4 lies completely outside the window, so it does not
requires clipping and is not saved.
 Line P5 to P6 lies inside and outside partially, so it requires clippi ng
and is clipped as shown in After Clipping of Fig 4.

4.3.1 Cyrus -Beck Line Clipping Algorithm :
Cyrus Beck Line clipping algorithm is a parametric line -clipping
algorithm. The term parametric means it is required to find the value of the
parameter ‘t’ in the parametric representation of the line segment for the
point at that the segment intersects the clipping edge.
 In Fig 5, XY is the line segment, which is intersecting at the two edges
A and B of the convex window.
 Apply the parametric equation of the line segment XY as:
X + t (Y -X)
where, 0 ≤ t ≤ 1 and t is the l inear parameter that continuously changes
the value
 Therefore,
X + t (Y -X) = (x1, y1) + t (( x2 – x1), (y2 – y1))
= (x, y) be any point on line XY
i. While t = 0, point X is obtained.
ii. While t = 1, point Y is obtained.
iii. While 0 ≤ t ≤ 1, line in between point X and Y is traced.
iv. t = 0.5, find s the midpoint of line XY.
v. While t < 0, trace the line on left hand side of X.
vi. While t > 0, trace the line on right hand side of Y.
 The variation in parameter ‘t’ generates line using pointwise method.
 The range of parameter values will identify the portion to be clipped
through any convex polygonal region consisting of n-vertices or lattice
points to be identified through the user.
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Fig 5 . Interaction of line XY with convex polygon window

Merits:
 It follows parametric approach.
 It is fast and efficient.

Demerits:

 The algorithm is can be applicable only for 2D line clipping.

4.3.2 Cohen -Sutherland Line Clipping Algorithm :

Cohen -Sutherland algorithm is a line -clipping algorithm that cuts lines to
portions, which are within a rectangular window.
 It removes the portion of the lines from a given set of lines and
rectangle area of interest also known as view port, which belongs
outside the area of interest and clip those lines that are partially inside
the area of interest.
 The algorithm divides a two -dimensional space into 9 regions (one
inside and eight outside) and then e fficiently determine the lines and
portions of lines that are visible in the central region of interest.
 Fig 6, defines the clipping regions for Cohen -Sutherland algorithm.
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Fig 6 lipping regions

 It uses 4 -bits to divide the entire region, and the centre region is the
clipping window.
 These 4 bits represents the TOP, BOTTOM, RIGHT and LEFT of the
region.
 In the Fig 6, the TOP and LEFT bit is set to 1 as it is in the TOP -
LEFT corner.
 There are three possibilities for the line:
o If the line is completely inside the window, the line is accepted.
o If the line is completely outside the window, the line is rejected
completely.
o If the line is partially inside the window, find the intersection point
and only draw that portion of the line that is inside the region.

Algorithm:
1. Assign a region code for each endpoint.
2. If both endpoints have a region code 0000 then accept this line.
3. Else perform the logical AND operation for both region codes
3.1 If the result is not 0000, then reject the line.
3.2 Else clipping is required.
3.2.1 Choose an endpoint of the line that is outside the window.
3.2.2 Find the intersection point at the window boundary based on
the region code.
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3.2.3 Replace the endpoint with the intersection point and update the
region code.
3.2.4 Repeat step 2 until we find a clipped line either trivially
accepted or trivially rejected.
4. Repeat step 1 for other lines.

Merits:
 This algorithm follows encoding approach.
 Easy to use

Demerits:
 This algorithm is applicable to rectangular window only and not to the
other convex shaped window.
 Less efficient.

4.3.3 Liang -Barsky Line Clipping Algorithm :

This algorithm is more efficient than Cohen -Sutherland line clipping
algorithm and can be extended to 3 -Dimensional clipping.
 It is a faster parametric line -clipping algorithm.
 It is based upon two concepts during clipping:
o It is based upon the parametric equation of the line.
o The intersections between the line and the clip window are
determined by the inequalities describing the range of the clipping
window.
 The parametric equation is given by:
x = x1 + t * dx
y = y1 + t * dy
where, dx = x2 – x1, dy = y2 – y1 and 0 ≤ t ≤ 1
 Liang -Barsky line -clipping algorithm uses four inequalities with two
parameters p and q, as defined in the algorithm below.

Algorithm:

1. Start
2. Read the two endpoints of the line as p 1 (x1, y1) and p 2 (x2, y2).
3. Read the two corners of the clipping window as xwmin, yw min, xw max,
ywmax.
4. Calculate the values of the parameters p i and q i for i = 1 to 4 such that:
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p3 = -dy, q3 = y1 – ywmin
p4 = dy, q4 = yw max – y1
5. If p i = 0, then the line is parallel to ith boundary.
If q i< 0, then :
The line is completely outside the boundary so discard the line.
Else:
Check whether line is horizontal or vertical and then check t he
line endpoints with the corresponding boundaries.
If the line endpoints lie within the bounded area then :
Use them to draw the line,
Else :
Use boundary coordinates to draw the line.
6. Initialize t 1 = 0 and t 2 = 1
7. Calculate q i / pi for i = 1 to 4
8. Select values of q i / pi where p i< 0 and assign maximum out of them
as t 1.
9. Select values of q i / pi where p i > 0 and assign minimum out of them
as t 2.
10. If (t1 < t2)
{
newx1 = x1 + t1 * dx
newx2 = x1 + t2 * dx
newy1 = y1 + t1 * dy
newy2 = y1 + t2 * dy
Draw line (newx1, newx2, newy1, newy2)
}
11. End

Merits:
 It uses the parametric approach.
 It is more efficient than any other line -clipping algorithms.

4.4 POLYGON CLIPPING
A shape with set of connected lines is a polygon.
 Polygons are clipped based on the window, the portion inside the
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 It clips the four edges in the boundary of the clip rectangle.
 The clip boundary determines the visible and invisible regions of
polygon c lipping and it is categorized in four ways:
o If visible region is fully inside the clip window, endpoints are saved.
o If visible region is fully outside the clip window, no endpoints are
saved.
o If visible exits the clip window, save the intersection.
o If the visible enters the clip window, save the endpoint and the
intersection.
 Each ed ge of the polygon must be tested against each edge of the clip
rectangle, new edges must be added and existing edges must be
discarded, retained or divided.
There are tw o types of polygons: (i) Concave (ii) Convex.
(i) Concave polygon :
It does not have any part of its diagonal in its exterior and at least one
angle should be greater than 180o.
(ii) C onvex polygon :

It has at least one part of diagonal in its exterior and all angles should be
less than 180o.


Fig 7 . Polygon Clipping

4.4.1 Sutherland -Hodgeman Polgygon Clipping :

Sutherland -Hodgeman polygon -clipping algorithm uses divide and
conquer strategy.
Procedure :
 It first clips the polygon against the right clipping boundary.
 The resulting polygon is then clipped against the top boundary.
 The same process is repeated for remaining two boundaries.
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 It works in order and makes sure that all edges of clipping polygon are
taken sequentially.
 At each step new sequence of ve rtices are generated and passed to next
window boundary for clipping.
 To clip the vertices following cases are considered:
Case 1- Both vertices are inside:
 Only the second vertex is added to the output list.
Case 2 - First vertex is outside while second one is inside:
 Both the point of intersection of the edge with the clip boundary and
the second vertex are added to the output list.
Case 3 - First vertex is inside while second one is outside:
 Only the point of intersection of the edge with the clip bou ndary is
added to the output list.
Case 4 - Both vertices are outside:
 No vertices are added to the output list.


Fig 8 . Before Clipping

Fig 9. After clipping using Sutherland -Hodgeman algorithm

Example :
 Fig. 9 depicts the following cases:
a. Clipping against the left side of the clip window.
b. Clipping against the top side of the clip window.
c. Clipping against the right side of the clip window.
d. Clipping against the bottom side of the clip window.


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Sutherland -Hodgeman comprises of two key processes in the algorithm:
o Perform inside -outside test to determine the visibility of a point or
vertex.
o Determine the intersection of the polygon edge and the clipping plane.
Method to determine the visibility of a vertex or a point .
 Let V be the vertex and AB the windo w boundary.
 Three points A, B and V define Vector AB and AV line in the plane .
 If this plane is considered in the xy plane, then the vector cross product
AV x AB has a z component given by:
(xv-xA)(yB-yA)-(yv-yA)(xB-xA)
 The sign of the z -component decides the position of point V with
respect to window boundary.
 It can be as follows:
o z>0: point is on the right side of window boundary
o z=0: point lies on the window edge
o z<0: point is on the left side of window boundary

Algorithm:

1. Read the coordinate of vertices of the subject polygon and clipping
polygon.
2. Consider an edge of the clipping window, compare the vertices of each
edge of the subject polygon with the clipping polygon or the edge, and
record the intersections.
3. Store the new intersection and v ertices in the new list of the vertices as
per the cases.
4. Perform steps 2 and 3 for the remaining edges of the clipping polygon
each time, the resulting list of polygon vertices are successively passed
to process the next edge of the clipping polygon.
5. Finish.

Merits:

 The process is simple and efficient.

Demerits:

 This method requires a considerable amount of memory.


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4.5 SUMMARY
 Cohen -Sutherland is easy to implement line-clipping algorithm, but it is
time consuming.
 Cyrus -Beck and Liang -Barsky uses parametric equation for clipping the
line.
 Liang -Barsky is faster and most efficient line clipping algorithm.
 Sutherland -Hodgeman is the most efficient algorithm for polygon
clipping. However, it requires more memory because new vertices are
obtained after clipping against each edge and it requires storing old and
new vertices.

4.6 UNIT END EXERCISE
1. Explain the Cohen -Sutherland line -clipping algorithm.
2. Explain the process of point clipping.
3. Write the algorithm for Sutherland -Hodgeman polygon clipping.
4. Explain the process for Cyrus -Beck line -clipping algorithm.
5. Write the algorithm for Liang -Barsky line -clipping algorithm.

4.7 REFERENCES
 D. Hearn and M. P. Baker, Computer Graphics , 2nd Edition, 1997.
 R. K. Maurya, Computer Graphics , John Wiley.

4.8 BIBLIOGRAPHY
1. J. Hughes, A. Van Dam, M. MCGuire, D. Sklar, J. Foley, and K.
Akeley, Computer Graphics, Principles and Practice , 3rd Edition,
Pearson, 2014.
2. P. Shirley and S. Marschner, “Fundamentals of Computer Graphics:
Ray Tracing,” CRC Press . p. 752, 2009.



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UNIT II

5

TWO DIMENSIONAL
TRANSFORMATIONS

Unit Structure
5.0 Objective
5.1 Introduction
5.2 Transformations and Matrices
5.2.1 Transformation
5.2.2 Matrices
5.3 Transformation conventions
5.3.1 2D coordinate system
5.3.2 3D coordinate system
5.4 2D transformations
5.4.1 Translation
5.4.2 Rotation
5.4.3 Scaling
5.4.4 Shearing
5.4.5 Reflection
5.5 Combined Transformation
5.6 Homogeneous Coordinates and Matrix Representation of 2D
Transformations
5.7 Transformation of Points
5.8 Transformation of the Unit square
5.9 Solid Body Transformations
5.10 Rotation about an Arbitrary Point
5.11 Reflection through an Arbitrary Line
5.12 The Window -to-Viewport Transformations
5.13 Summary
5.14 Unit End Exercise
5.15 References

5.0 OBJECTIVE
This chapter will able you to understand the following concept :
 Transformations and Matrices with its representation
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 Methods of transformation
 Ways of Transformation about an Arbitrary point
 Transformation of homogeneous coordinates
 Window -to-Viewport Transformations.

5.1 INTRODUCTION
To convert the real world object into computer we need to understand the
coordinate system. The real world objects are represent ed in computer
with the different angles and different perspective with the help of
different coordinate system . The concept of transformation is used to
implement this process. Transformation takes place in 2 ways that are 2D
transformation and 3D transform ation.

In this chapter we will see the 2D coordinate system of converting real life
object to computer.

5.2 TRANSFORMATIONS AND MATRICES
5.2.1 Transformation :

Creation of graphs and images in computer are not sufficient in computer
graphics. In real world object are moved from one position to another or
its changing its shape or or ientation and it can be seen fro m different
angles and different perspective.
The process of changing size, shape, position and orientation of object is
called transformation.
This change can be done by y
1. Geometric transformation:
In this type of transformation the object
change its appearance with the help
Geometric properties of the specif ied
points. x

Fig. 5 .2.1
2. Coordinate transformation : y
In this type of transformation the
object is y’ appeared with
the modified points on the screen
after performing some non -singular
x’ operation like translation,
rotation, scaling etc.


x
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5.2.2 Matrices:

 It provides the technique to represent the real world object into
computer system.
 It also provides a way to calculate different type of geometric function
on the real life object to show on the screen.
 With help of matrices various operation like translation scaling etc
can be done on the objec t
 General Matrix representation of 3 ×3 matrix ൥143
598
824൩

 Homogeneous Matrix representation of 3 ×3 matrix ൥100
010
001൩

5.3 TRANSFORMATION CONVENTIONS
To draw an object onto screen, computer need the system that is called as
conventions of transformation. This can be done with 2 types such as 2D
coordinate system and 3D coordinate system.
For 2D image we use 2D coordinate system as convention for
transformation whereas for 3D image we use 3D coordinate system as
convention for transformation of image.

5.3.1 2D COORDINATESYSTEM:
This 2D coordinate system consist s of two coordinates x and y i.e. x plane
and y plane . Its direction is along with the x and y axis with reference to
an origin. With the help of this system the real word object can be draw on
screen with lines, circle, polygon etc
Y Z X
Fig 5.3.1

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5.3.2 3D coordinate system:
This 3D coordinate system consist s of three coordinates x, y and z .i.e. x
plane, y plane and z plane.
Its direction is along with the x, y and z axis with reference to an origin.
With the help of this system the real word object can be draw n on screen
with lines, circle, polygon etc.

Rules applied for transformation y
Y X Z Fig 5.3. 2

1. Right hand coordinate system :
 In this coordinate system right hand figures
( X-Thumb, Y-index and X-middle)
used as axis to represent the position of the transformed image .
 In Right hand coordinate system all coordinates are positive so the
object will transform
in pos itive direction only.
origin(0, 0, 0)











Fig 5.3.3
Left hand coordinate system :
 In this coordinate system right hand figures( X -Thumb, Y -index and
Z-middle) used as axis to represent the position of the transformed
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 In Right hand coordinate system x and y coordinates are positive and
z coordinate is negative so the object will transform in both the
direction .

5.4 2D TRANSFORMATIONS
The transformation which is applied on 2 plane i.e. x and y plane to
change the appearance of the object is called 2D transformation.
Basic 2D transformation 1. Translation Denoted by T 2. Rotation Denoted by R 3. Scaling Denoted by S 4. Shearing Denoted by Sh 5. Reflection Denoted by RE
5.4.1 Translation :
 Changing the position of an object or shifting of an object is called 2D
translation.
 To achieve translation an object should shift in x and y direction.
 We can translate a two dimensional figure by adding translation
factor t x and t y to the original points x and y
 The following matrix shows translation, P=P+T
𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ+𝑇ቂ tx
ty ቃ

Example: Apply translation transformation on BC, with A=(2,5), B=(2 ,1),
C=(5,1) with translation factor x=2, y=2.
Solution: 𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩+𝑇൥1 0 0
0 1 0
2 2 1൩

𝑃 ൥𝐴
𝐵
C൩=𝑃൥3 5 0
2 2 0
7 3 2൩=൥3 5
2 2
7 3൩
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Fig 5. 4.1 Translation

5.4.2 Rotation :
 Changing the position of an object with rotation angle along with xy
plane is called 2D rotation.
 To achieve rotation an object should move in x and y direction with
the specified angle  .
 We can rotate a two dimensional figure by multiplying rotation
factor rx and ry to the original points x and y
 The following matrix shows rotation , P=P+R
Clockwise rotation 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ. 𝑅ቂcos
sin −𝑠𝑖𝑛
𝑐𝑜𝑠ቃ

Anti Clockwise rotation 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ. 𝑅ቂcos
−sin 𝑠𝑖𝑛
𝑐𝑜𝑠ቃ

Example: Apply rotation transformation on ABC, with A=(2,5),
B=(2,1), C=(5,1) with the rotation angle 90° in clockwise direction.
𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅቎cos
−soin
0
𝑠𝑖𝑛 0
𝑐𝑜𝑠 0
0 1 ቏
𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅൥010
−100
001൩ = 𝑃൥−2 2 0
−1 2 0
−1 5 1൩
by putting cos90 = 0 and sin90 =1
A=(-5, 2) B =(-1, 2) C =(-1,5)
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Fig 5.4.2 Rotation
5.4.3 Scaling :
 Changing the size of an object is called 2D scaling .
 To achieve scaling an object should change the size of object in x and
y direction.
 We can scale a two dimensional figure by multiplying scaling factor
sx and sy to the original points x and y
The following matrix shows scaling , P=P+S
 Scaling in x and y direction 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆൤ 𝐬𝐱 𝟎
0 𝐬𝐲൨
 Scaling in x direction


 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆ቂ 𝐬𝐱 𝟎
0 𝟏ቃ
Scaling in y direction 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆൤ 𝟏 𝟎
0 𝐬𝐲൨
Example: Apply scaling transformation on ABC, with A=(2,5),
B=(2,1), C=(5,1) with the scaling factor x=2 and y=2
Solution: Scaling in x and y direction 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆൤ 𝐬𝐱 𝟎
0 𝐬𝐲൨

𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅൥2 0 0
0 2 0
0 0 1൩ = ൥4 10 0
4 2 0
10 2 1൩ by substituting original
value of point A,B,C and scaling factor x=2, y=2
A=(4, 10) B =(-4, 12) C =(-10, 2)
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Fig 5.4.3 Scaling
5.4.4 Shearing:

 A shear is something that pushes the object sideway and make a thing
little bit tilled.
 To achieve shearing an object should tilled(slant) with respect to x and
y direction.
 We can shear a two dimensional figure by multiplying shearing factor
shx and s hy to the original points x a nd y
The following matrix shows shearing , P=P+S h
𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆ቂ 𝟎 𝐬𝐡𝐲
𝐬𝐡𝐱 0 ቃ

Shea𝑟𝑖𝑛𝑔 𝑤𝑖𝑡h 𝑥 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆hxቂ𝟏 𝟎
𝐬𝐡𝐱 1 ቃ

𝑆h𝑒𝑎𝑟𝑖𝑛𝑔 𝑤𝑖𝑡h 𝑦 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆hyቂ 𝟏 𝐬𝐡𝐲
𝟎 1 ቃ

Example: Apply shearing transformation on ABC, with A=(2,5),
B=(2,1), C=(5,1) with the shearing factor x=2 and y=2 separately .

Solution: Shearing in x direction 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆ቂ 𝟏 𝟎
Shx 𝟏ቃ

𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅൥1 0 0
2 1 0
0 0 1൩ = ൥12 5 0
4 1 0
6 1 1൩ by substituting

original value of poin t A,B,C and shearing factor x=2
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Fig 5. 4.4.1 Shearing with X axis

Shearing in y direction 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑆ቂ 𝟏 𝐬𝐡𝐲
𝟎 1 ቃ

𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅൥1 2 0
0 1 0
0 0 1൩ = ൥2 9 0
2 5 0
5 11 1൩ by substituting original

value of point A,B,C and shearing factor y=2

Fig 5.4.4.2 Shearing with Y axis

5.4.5 Reflection:

 Creating mirror image of an object is called 2D reflection .
 To achieve reflection an object should appear in the form of mirror
object in x and y direction.
 We can reflect a two dimensional figure by multiplying reflection
factor REx and REy to the original points x and y
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 The following matrix shows reflection , P=P+RExy
Reflection 𝑤𝑖𝑡h 𝑥𝑦 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑅𝐸xቂ−𝟏 𝟎
𝟎 −𝟏ቃ
Reflection 𝑤𝑖𝑡h 𝑥 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑅𝐸xቂ𝟏 𝟎
𝟎 −𝟏ቃ
𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡h 𝑦 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑅𝐸yቂ−𝟏 𝟎
𝟎 𝟏ቃ

Example: Apply reflection transformation on  ABC, with A=(2,5),
B=(2,1), C=(5,1)
Solution:
Reflection 𝑤𝑖𝑡h 𝑥𝑦 𝑎𝑥𝑖𝑠 𝑃 ൤𝑥
𝑦൨=𝑃ቂ𝑥
𝑦ቃ.𝑅𝐸xቂ−𝟏 𝟎
𝟎 −𝟏ቃ
𝑃 ൥𝐴
𝐵
C൩=𝑃൥2 5 0
2 1 0
5 1 1൩ . 𝑅൥−1 0 0
0 −1 0
0 0 1൩ = ൥−2 −5 0
−2 −1 0
−5 −1 1൩ by

substituting original value of point A,B,C and scaling matrix


Fig 5. 4.5 Reflection

5.5 COMBINED TRANSFORMATION
Some time an object may require more than one transformation to get
desired output, we need to transform an object at several times with
different transformation like rotation, scaling, translation, shearing,
reflection such a group of transformation is called combined
transformation.

The sequence of transformation is applied on an object to get the required
shape of object.
It is also called composite transformation.
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Method:
When we need to apply more than one transformation on an object first we
need to do single transformation in matrix form the we need to apply
second transformation on an object in matrix form then the final matrix
multiplication is considered for the desired object transformation.

Example: Apply combined transformation on ABC, with A=(2,5),
B=(2,1) , C=(5,1) apply the following sequence of transformation on
ABC

a) Rotation by angle =90
b) Reflection through x axis
Solution: The rotation matrix =90 ref fig 5.4.4

[R]= ൥cos 90 sin 90
−sin90 cos90
൩= ቂ𝟎 𝟏
−𝟏 0ቃ
The reflection through x axis matrix
[REx]= ቂ1 0
0−1ቃ

The combined transformation matrix is
[R].[RE x]= ቂ𝟎 𝟏
−𝟏 0ቃ. ቂ1 0
0−1ቃ= . ቂ0 −1
−1 0ቃ

Apply combined transformation on original points
[P]=൥𝐴
𝐵
𝐶൩.([R].[REx])= ൥2 5
2 1
5 1൩. ቂ0 −1
−1 0ቃ

[P]=൥𝐴
𝐵
𝐶൩= ൥−2−5
−2−1
−5−1൩

5.6 HOMOGENEOUS COORDINATES AND MATRIX REPRESENTATION OF 2D TRANSFORMATIONS
Translation : ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥1 0 𝑇𝑥
0 1 𝑇𝑦
0 0 1 ൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

Rotation : ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥𝑐𝑜𝑠 −𝑠𝑖𝑛 0
𝑠𝑖𝑛 𝑐𝑜𝑠 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
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Scaling : ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥𝑆𝑥 0 0
0 𝑆𝑦 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

Shearing in x direction: ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥1 0 0
𝑆𝐻𝑥 1 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

Shearing in y direction: ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥ 1 𝑆𝐻𝑦 0
0 1 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

Reflection in x direction : ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥1 0 0
0−1 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

Reflection in y direction : ൥𝑋 𝑛𝑒𝑤
𝑦 𝑛𝑒𝑤
1൩=൥−1 0 0
0 1 0
0 0 1൩ . ൥𝑋 𝑜𝑙𝑑
𝑌 𝑜𝑙𝑑
1൩

5.7 TRANSFORMATION OF POINTS

All types of transformation such as translation, rotation, scaling, reflection,
shearing is applied to get the transformation of point. All the
transformation elements can be described through matrices to perform
transformation on points.









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Fig 5.7 .1 Transformation of point

5.8 TRANSFORMATION OF THE UNIT SQUARE
When we transform an object in 2D transformation , all the points are
transformed except the origin point it will remain unchanged , which
indicates that the conversion of an object in to new shape in 2D plane with
xy coordinates. These points are considered as apposition vectors of a unit
square with one point situated at the origin of the coordinate system. As
we can see in fig 5. 8.1 the origin coordinate remains same i.e. A and A 
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are same but the rest of the coordinates are changes according to the unit
vector , hence the transformation is applied on all point except origin
point.
The general form of transformation matrix is [p ]= [p] [T]
[x y]= [x y] . ቂ𝑎 𝑏
𝑐 𝑑ቃ

Fig 5. 8.1 Unit square transformation

5.9 SOLID BODY TRANSFORMATIONS
 Solid body transformation is based on vector transformation. Before
performing transformation basic physics quantities it should be
known.
 We know that in solid body transformation parallel lines are
transformed as parallel lines but it is not necessary that perpendicular
line will always transformed perpendicular only.
 Angle between lines are also not preserved.
 To preserve an angle between two intersecting line solid body
transformation is used. To achieve this, dot product and dot product of
vector are required.

Vector: A vector is quantity which has direction as well as magnitude. It
is used to determine the position of one point in space to another. Vector is
used in graphics also known as vector graphics. Vector graphics is the
creation of digital image through sequence of commands of mathematical
statements that places lines and shapes i n given two dimensional or three
dimensional space. Vector graphics describes series of points to be
connected.

Dot Product of vector: they can be multiplied using dot product of matrix
is written using central dot present between these two vector.

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Cross Product of vector: the cross product of two vector is an another
vector which is right angled to both the vector like a x b

5.10 ROTATION ABOUT AN ARBITRARY POINT
In translation and homogeneous coordinate system we have seen that the
homogeneous coordinate system accomplishes the rotation about the
points of an origin whereas, 2D transformation does not allow rotation of
points at origin.
General matrix of homogeneous matrix for translation ൥1 0 0
0 1 0
𝑚 𝑛 1൩

When the translation is about origin then m will be changed as –m and n
will be changed as –n

Homogeneous matrix for translation about an origin ൥1 0 0
0 1 0
−𝑚 −𝑛 1൩

Steps to achieve Rotation about arbitrary points
1. The points of translation about the origin m= -m and n= -n
2. Perform specified rotation.
3. Translate result back to original points (center of origin)
4.
[𝑥,𝑦,𝑧]=൥1 0 0
0 1 0
−𝑚 −𝑛 1൩൥𝑐𝑜𝑠 𝑠𝑖𝑛 0
−𝑠𝑖𝑛 𝑐𝑜𝑠 0
0 0 1൩൥1 0 0
0 1 0
𝑚 𝑛 1൩

Example: Center of an object is at point (5,6) and the object is rotated
about 90  counter clockwise about its origin using the matrix of rotation.
൥1 0 0
−1 0 0
0 0 1൩

Solution: translating the point to the origin i.e m= -m and n= -n;
origin (m,n)m=5 and n=6
[𝑥,𝑦,𝑧]=൥1 0 0
0 1 0
−𝑚 −𝑛 1൩൥𝑐𝑜𝑠 𝑠𝑖𝑛 0
−𝑠𝑖𝑛 𝑐𝑜𝑠 0
0 0 1൩൥1 0 0
0 1 0
𝑚 𝑛 1൩
Origin matrix rotation
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[𝑥,𝑦,𝑧]=൥1 0 0
0 1 0
−5 −6 1൩൥0 1 0
−1 0 0
0 0 1൩൥1 0 0
0 1 0
5 6 1൩

[𝑥,𝑦,𝑧]=൥0 1 0
−1 0 0
6 −5 1൩൥1 0 0
0 1 0
5 6 1൩
[𝑥,𝑦,𝑧]=൥1 0 0
0 1 0
11 6 1൩

5.11 REFLECTION THROUGH AN ARBITRARY LINE
Reflection is achieved by mirroring the line coordinates on x axis, y axis
and even by passing line through origin when reflection is taken place on
both x and y axis.

We know that when x and y both the coordinates are changed, the line is
reflected and passes through the origin. But sometimes reflection of an
object through a line which does not pass through an origin id required.

Step 1: Translation: translat e the line and the object so that the line
passes through origin. [T]

Step 2: Rotation: Rotate the line and the object the origin until the line is
coinciding with one of the coordinate axis. [R]

Step 3: Reflection: reflect through the coordinate axis [R]

Step 4: Inverse rotation: apply inverse rotation about the origin. [R]-1

Step 5: Inverse translation: translate back to original location. [T]-1

5.12 THE WIN DOW – TO - VIEWPORT TRANSFORMATIONS
A world coordinate area selected for display is called a window . An area
on display device to which a window is mapped is called viewport. The
window defines what to be display on screen whereas, the viewport
defines where is to be displayed. The standard rectangle position is applied
to see window and viewport . The part of real word image (world
coordinate system) seen on device coordinate system (screen coordinate
system) is called as viewing transformation. Sometimes two dimensional
transformations is also called as window to viewport transformation .

In computer graphics, the term window is referred to an area of image that
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image on screen the viewing coordinate is rectangular wind ow which
shows the enlarge image in viewport. By changing the position of the
viewport, we can view object at different position on the display area of
output device. Also, by varying the size of viewports, we can change the
size and proportions of display ed objects. When all coordinate
transformations are completed, viewport clipping can be performed in
normalized coordinates or in device coordinates. Viewport is typically
defines in unit square.

Process of window to viewport coordinate system :






Fig 5.1 2.1 window to viewport system

Window to viewport mapping:

A point at position (xw, yw) in the window is mapped into new position
(xv , yv) in associated view port .

Fig 5.1 2.2 Window to Viewport transformation

5.13 SUMMARY
The process of changing size, shape, position and orientation of object is
called transformation. The basic transformation is translation, rotation,
reflection , scaling , shearing. When the object is moved from one position
to another position it is called as translation.hen we rotate an object in
180 it is called as rotation whereas when we create mirror image of object
it is called as reflection. When we change the shape of the object is called
as scaling whereas when we create tiled shape of any object i t is called as
shearing. We can achieve transformation by using multiplication and
addition function on points given. Through the transformation we can
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change the appearance of the object.wew can use 3 X 3 matrixes for 2D
transformation whereas 4 X 4 matri xes for 3D transformation. We can get
transformation through x and y axis in 2D transformation. We can get
transformation through x, y and z axis in 3D transformation. Using more
than one transformation we can combined transformation to achieve
multiple tr ansformation. We can represent transformations in
homogeneous coordinate system.

5.14 UNIT END EXERCISE
1. What is transformation? Explain its types.
2. Explain 2D transformation in detail.
3. Explain homogeneous coordinate system in detail.
4. What is combined transformation? Explain in detail.
5. How to convert an object from window to viewport?
6. Write a short note on rotation about an arbitrary point.
7. Explain in brief solid body transformation.
8. Write a short note on unit square transformation.
9. Explain reflection through an arbitrary line.
10. Explain the concept of transformation of points.

5.15 REFERENCES
 Computer Graphics C version 2nd Edition by Donald D. Hearn and M.
Pauline Baker
 Computer Graphics A programming approach 2nd Edition by Steven
Harrington McGraw Hill
 Fundamental of Computer Graphics 3rd Edition by Peter Shirley an
Steven Marschner


*****


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UNIT II

6

THREE DIMENSIONAL
TRANSFORMATIONS

Unit Structure
6.0 Objective
6.1 Introduction
6.2 Transformations and Matrices
6.2.1 Transformation
6.2.2 Matrices
6.3 Transformation conventions
6.3.1 3D coordinate system
6.4 3D transformations
6.4.1 Translation
6.4.2 Rotation
6.4.3 Scaling
6.4.4 Shearing
6.4.5 Reflection
6.5 Multiple Transformations
6.5.1 Method
6.6 Homogeneous Coordinates and Matrix Representation of 3D
Transformations
6.7 Rotation about an arbitrary axis in space
6.8 Reflection through an arbitrary plane
6.9 Affine and Perspective Geometry
6.10 Perspective Transformations
6.11 Techniques for Generating Perspective Views
6.12 Axonometric Projection
6.13 Oblique Projection
6.14 Orthographic Projection
6.15 Vanishing Points
6.16 View volumes for projections
6.17 Summary
6.18 Unit End Exercise
6.19 References
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6.0 OBJECTIVE
 This chapter will able you to understand the following concept
 3D Transformations and Matrices with its representation
 Types of transformation such as scaling, reflection, translation,
shearing and rotation.
 Ways of Transformation about an Arbitrary plane and arbitrary axis.
 Types of projection such as Orthographic, Axonometric, Oblique
Projections.
 Camera model and its perspective geometry

6.1 INTRODUCTION
When we want to convert the real life object into computer we
need to understand the coordinate system. The real life objects represent in
computer with the different angles and different perspective with the help
of different coordinate system and here the concept of transformation
exist. Transformation takes place in x axis, y axis, z axis in 3D
transformation. In this chapter we will see the 3D coordinate system of
converting r eal life object to computer. The transformation which takes
place on 3D plane or 3D space is called as 3D transformations.

6.2 TRANSFORMATIONS AND MATRICES
6.2.1 Transformation :
Creation of graphs and images in computer are not sufficient in
computer graphics. In real world object s are moved from one position to
another or its changing its shape or orientation and it can be seen from
different angles and different perspective.

The process of changing size, shape, position and orientation of object is
called transformation.

This change can be done by Geometric transformation:
In this type of transformation the object changes its appearance with the
help Geometric properties of the specified points.





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1. Coordinate transformation:
In this type of transformation the object y’ appears with the
modified points on the screen after performing some non -singular x’
operation like translation, rotation, scaling etc.




6.2.2 Matrices:
 In computer graphics, objects can be described in various quantitative
number s also called points. To represent these quantities, matrices are
used.
 It provides the technique to represent the real world object into
computer system.
 It also provides a way to calculate different type of geometric function
on the real life object to show on the screen.
 With help of matric es various operation like translation scaling etc
can be done on the object
 General Matrix representation of 3 ×3 matrix ൥143
598
824൩
 Homogeneous Matrix representation of 3 ×3 matrix ൥100
010
001൩

6.3 TRANSFORMATION CONVENTIONS
To draw an object onto screen, computer need the system that is
called as conventions of transformation. This can be done with 2 types
such as 2D coordinate system and 3D coordinate system.

For 2D image we use 2D coordinate system as convention for
transformation whereas for 3D image we use 3D coordinate system as
convention for transformation of image.

6.3.1 3D coordinate system:
 This system is made up of three coordinates: x, y, and z,
which correspond to the x, y, and z planes, respectively and i ts
direction along with the x, y and z axis with reference to an origin.
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With the help of this system, the real word object can be drawn on
screen with lines, circles, polygons, etc. Rules applied for
transformation

fig 6.3.1

1. Right hand coordinate system :
 In this coordinate system right hand figures ( X -Thumb, Y -index and
X-middle) used as axis to represent the position of the transformed
image.
 In Right hand coordinate system all coordinates are positive so the
object will transform x in positive direction only.



fig 6.3.2

2. Left hand coordinate system :
 In this coordinate system right hand figures( X -Thumb, Y -index and
Z-middle) used as axis to represent the position of the transformed
image.

Fig 6.3.3
 In Right hand coordinate system x and y coordinates are positive and
z coordinate is negative so the object will transform in both the
direction.


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6.4 3D TRANSFORMATIONS
The tran sformation which is applied on 3 plane i.e. x , y and y z plane to
change the appe arance of the object is called 3 D transformation.
Basic 3D transformation 1. Translation Denoted by T 2. Rotation Denoted by R 3. Scaling Denoted by S 4. Shearing Denoted by Sh 5. Reflection Denoted by RE`
6.4.1 Translation
 Changing the position of an object or shifting of an object is called 3 D
translation.
 To achieve translation , an ob ject should shift in the x, y, and z
direction.
 We can translate a two dimensional figure by adding translation factor
tx, ty and t z to the original points x , y and z .
 The following matrix shows translation, P =P+T

𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉+𝑇ቈ tx
ty
tz ቉

General representation of 3D translation matrix
𝑃 ൥𝐴
𝐵
C൩=𝑃 [𝑎,𝑏,𝑐,1]+𝑇൦1 0 0 0
0 1 0 0
0 0 0 1
𝑙 𝑚 𝑛 1൪
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Fig 6.4 .1 3D Translation

6.4.2 Rotation :
 Changing the position of an object with rotation angle along with xyz
plane is called 3 D rotation.
 To achieve rotation an object should move in x and y direction with
the specified angle  .
 We can rotate a two dimensional figure by multiplying rotation
factor r x , ry and r z to the original points x, y and z
 The following matrix shows rotation, P =P+R

Rotation 𝑎𝑏𝑜𝑢𝑡 𝑥 𝑎𝑥𝑖𝑠 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉. 𝑅൥1 0 0
0 𝑐𝑜𝑠 𝑠𝑖𝑛
0 −𝑠𝑖𝑛 𝑐𝑜𝑠൩

Rotation 𝑎𝑏𝑜𝑢𝑡 𝑦 𝑎𝑥𝑖𝑠 𝑃൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉. 𝑅൥𝑐𝑜𝑠 0 –𝑠𝑖𝑛
0 1 0
𝑠𝑖𝑛 0 𝑐𝑜𝑠൩

Rotation 𝑎𝑏𝑜𝑢𝑡 𝑧 𝑎𝑥𝑖𝑠 𝑃൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉. 𝑅൥𝑐𝑜𝑠 𝑠𝑖𝑛 0
−𝑠𝑖𝑛 𝑐𝑜𝑠 0
0 0 1൩
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Fig 6 .4.2 3D Rotation

6.4.3 Scaling :
 Changing the size of an object is called 3D scaling.
 To achieve scaling an object should change the size of object in x, y
and z direction.
 We can scale a two dimensional figure by multiplying scaling factor
sx , sy and s z to the original points x, y and z
 There are two types of scaling
1 Uniform scaling: if all scaling factors are same then it is called as
Uniform Scaling
2 Non-uniform scaling: if all scaling factors are different then it is
called as Non - Uniform Scaling
 The following matrix shows scaling, P=P+S

Scaling x, y and z direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൥𝑆𝑥00
0𝑆𝑦0
00𝑆𝑧൩

Uniform Scaling x, y and z direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൥100
010
001൩

Non-Uniform Scaling x, y and z direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൥100
020
003൩


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Fig 6.4.3 3D Scaling

6.4.4 Shearing :
 A shear is something that pushes the object sideway and make a thing
little bit tilled.
 To achieve shearing an object should tilled(slant) with respect to x, y
and z direction.
 We can shear a two dimensional figure by multiplying shearing
factor sh x , shy and sh z to the original points x, y and z.
 The following matrix shows shearing, P =P+Sh

Shearing x, y and z direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 𝑎 𝑏 0
𝑐 1 𝑑 0
𝑒 𝑓 1 0
0 0 0 1൪

Shearing in xy direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 0 0 0
0 1 0 0
𝑎 𝑏 1 0
0 0 0 1൪

Shearing in yz direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 𝑎 𝑏 0
0 1 0 0
0 0 1 0
0 0 0 1൪

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Shearing in xz direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 0 0 0
𝑎 1 𝑏 0
0 0 1 0
0 0 0 1൪


Fig 6.4.4 3D Shearing

6.4.5 Reflection :
 Creating mirror image of an object is called 3 D reflection.
 To achieve reflection , an object should appear in the form of mirror
object in x, y and z direction.
 We can reflect a two dimensional figure by multiplying reflection
factor RE x , RE y and RE z to the original points x, y and z
 The following matrix shows reflection, P =P+RExy

Reflection xy direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 1൪

Reflection xz direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 1൪

Reflection yz direction 𝑃 ൥𝑥
𝑦
z൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆 ൦−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1൪

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Reflection about origin p ൥𝑥
𝑦
𝑧൩=𝑃ቈ𝑥
𝑦
𝑧቉.𝑆൦−1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1൪



Fig 6.4.5 3D Reflection

6.5 MULTIPLE TRANSFORMATION
Sometime an object may require more than one transformation to
get desired output, we need to transform an object at several times with
different transformation like rotation, scaling, translation, and shearing ,
reflection such a group of transformation is called combined
transformation.

Multiple transformation means that applying more than one
transformation on single object by using respective transformation matrix.
The sequence of transformation is applied on an object to get the requir ed
shape of object.
6.5.1 Method:
When we need to apply more than one transformation on an object
first we need to do single transformation in matrix form the we need to
apply second transformation on an object in matrix form then the final
matrix multipl ication is considered for the desired object transformation.

Example: Apply combined transformation on ABC, with A = (2, 5), B=
(2, 1), C= (5, 1) apply the following sequence of transformation on
ABC
a) Rotation by angle =90
b) Reflection through x axis

Solution: The rotation matrix =90 ref fig 5.4.4

[R]= ൥cos 90 sin 90
−sin90 cos90
൩= ቂ0 1
−1 0ቃ
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The reflection through x axis matrix
[RE x]= ቂ1 0
0−1ቃ
The combined transformation matrix is

[R].[RE x]= ቂ0 1
−1 0ቃ. ቂ1 0
0−1ቃ= . ቂ0 −1
−1 0ቃ

Apply combined transformation on original points

[P]=൥𝐴
𝐵
𝐶൩.([R].[REx])= ൥2 5
2 1
5 1൩. ቂ0 −1
−1 0ቃ

[P]=൥𝐴
𝐵
𝐶൩= ൥−2−5
−2−1
−5−1൩

6.6 HOMOGENEOUS COORDINATES AND MATRIX REPRESENTATION OF 3D TRANSFORMATIONS
In 2D coordinate system homogeneous coordinate can be represented in 3
X 3 matrix whereas, in 3D coordinate system homogeneous coordinate can
be represented in 4 X 34 matrix.

Hence instead of only x, y, z in 3D coordinate system we could use x, y, z,
h where h ≠ 0
Translation: [T] = ൦1 0 0 0
0 1 0 0
0 0 1 0
𝑙 𝑚 𝑛 1൪

Rotation: [Rx] =൥100
0cossin
0−sincos൩

Rotation: [Ry] =൥cos0−sin
010
sin0cos൩


Rotation: [Rz] =൥cossin0
−sincos0
001 ൩

Scaling: [𝑆]=൥𝑆𝑥00
0𝑆𝑦0
00𝑆𝑧൩
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Shearing: [𝑆h]=൦1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1 ൪

Reflection: [𝑅𝐸.𝑥]=൦−1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1൪

6.7 ROTATION ABOUT AN ARBITRARY AXIS IN SPACE
Rotation about an arbitrary axis is implemented using translation and
rotation about the coordinate axis. Hence there should be coordinate axis
which is coincident to the arbitrary rotation axis. In 3D we have ro tation
about an arbitrary point through coordinate axis which is rotate around
itself of the object

Steps to achieve Rotation about an arbitrary Axis in space

1. The origin point which is to be translate p=[a x, by, cz] trough all 3 axis

[T]= ൦1 0 0 0
0 1 0 0
0 0 1 0
−ax −by−cz 1൪

2. Perform specified rotation about an arbitrary axis x and y. Rotation in
anticlockwise angle implies positive direction therefore value of  is
positive

Rotation about an arbitrary x axis (anticlockwise rotation)
[Rx] =൥100
0cossin
0−sincos
൩

Rotation about an arbitra ry y axis ( clockwise rotation)

[Ry] =൥cos0sin
010
−sin0cos൩
3. Rotation about an angle  in z axis [Rz] =൥cossin0
−sincos0
001൩
4. Apply Inverse rotation on step 2

5. Inverse Translation of step 1 to make an arbitrary axis in a particular
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6.8 REFLECTION THROUGH AN ARBITRARY PLANE
Reflection is achieved by mirroring the line coordinates on x axis, y axis
and even by passing line through origin when reflection is taken place on
both x and y axis.

We know that when x and y both the coordinates are changed; the line is
reflected and passes through the origin. But sometimes refle ction of an
object through a line which does not pass through an origin id required.
Step 1: Translation: translate the point P which is lies on reflection plane .
[T]
Step 2: Rotation: Rotate the point about x and y axis . [R]
Step 3: Reflection: reflect an object through z axis or xy plane with z=0
[R]
Step 4: Inverse rotation: apply inverse rotation about 2 axes which state in
step 2 . [R]-1
Step 5: Inverse translation: perform inverse translation of step 1 to get
back to original location. [T]-1

6.9 AFFINE AND PERSPECTIVE GEOMETRY
 Study of parallel lines in an object is called affine geometry.
 In affine geometry lines of object are always parallel whereas in
perspective geometry lines of object are always perpendicular means
not parallel.
 Affine transformation can be achieved by combining translation,
scaling, rotation and shearing together on object.
 In affine geometry study of connected lines are concerned.
 To convert one object shape into another shape in three dimensions
affine and pe rspective transformation is used.
 When 3D object is transformed into 2D object it is called as
projection.
 In perspective projection the object is projected by center of
projection of 3D image on 2d plane. If the center of projection is
located at a fini te point in 3D space then it is called as perspective
projection.
 Another type of projection in which the center of projection has been
move to infinite is called parallel projection.
 In parallel projection the information of the origin object remains
same and the maintain realistic view of an image.

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A1 A2 B1 B2 C1 C2

Fig 6.8.1 affine transformation and geometry

6.10 PERSPECTIVE TRANSFORMATIONS
When human eyes see far away things they are looking smaller as
they come closer they are looking bigger to eyes this type of viewing is
called perspective. This technique need to convert an object from one form
to another and that is known as projection. Perspective projectio n will
convert 3D image into 2D as human eyes does . The projection transform s
shape of object into another form as they appear close to eyes. Perspective
transformation represented in homogeneous coordinate system is as
follows in 4 X 4 matrix [PT]=൦1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1൪ where r is any non zero
value

There are three types of perspective transformation
1. Single point perspective transformation
2. Two point perspective transformation
3. Three point perspective transformation

1. Single point perspective transformation : If one term (either x, y, z)
in fourth column of matrix transformation is non zero we will get single
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Fig 6.10.1 Single point perspective transformation

1. Two point perspective transformation: If two term (either x, y, z) in
fourth column of matrix transformation is non zero we will get single
point perspective transformation. In this type object can be seen in two
directions.

Fig 6.10.2 Two point perspective transformation

2. Three point perspective transformation: If three term (either x, y, z)
in fourth column of matrix transformation is non zero we will get single
point perspective transformation. In this type object can be seen in three
directions.
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Fig 6.10.3 Three point perspective transformation

6.11 TECHNIQUES FOR GENERATING PERSPECTIVE VIEWS
Translation and rotation operations are used to create perspective view
of an object. When we see object from multiple directions, the object
makes 3D view. For a fixed center of projection, a single point perspective
projection is carried out by multiple faces (One, two a nd three
perspective). Ref fig 6.10.1, 6.10.2, 6. 10.3
1. Translation of object
2. Rotation

6.12 AXONOMETRIC PROJECTION
The problem of rendering for two -dimensional view of
dimensional object existed long before the computer was used. One class
of projection often used in drafting is called an axonometric projection .
This is a parallel projection for which the direction of projection is
perpendicular to the view plane. We can alter the direction of an
axonometric projection to get different view of an object , provided we
also change the view plane normal to match the projection direction. If we
change our direction slightly to the side , then one of the side faces will
become visible , while the edges of the front face will shorten. If we raise
our angle of view top edges lengthen make the top face visible, while the
edges on the side faces appear to shrink.

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Types of axonometr ic projection are
1. Isometric projection
2. Dimetric projection
3. Trimetric Projection

1. Isometric projection : It is mai nly used in engineering drawing. In this
type of pro jection the object is viewing from all the direction is looking
same and the principal axis are equally distributed on same distance with
common angle of 120  among the entire axis. Because of the equal angle
among all three axis the distance and the length of the all side of object
will be same . All The parallel lines are equal in length .


Fig 6.12.1 Isometric Projection

2. Dimetric projection : In isometric projection all three axis appears to
be equally distanced but in Diemetric projection instead of all three, any
two axis are equally distanced and one axis is different distanced than
other two. This projection says that the view created through principal
axis are represented the angle of object. Since two axis ar e same the line
which are drawn along axis are also same.
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Fig 6.1 2.2 Dimetric Projection

3. Trimetric Projection : isometric projection all three axis appears to be
not equally distanced. All three axis are unequally distanced as well as the
angles created by these lines are also unequal.


Fig 6.1 2.3 Trimetric Projection

6.13 OBLIQUE PROJECTION
If the direction of parallel projection is not parallel to view plane then we
have oblique projection . There are two types of oblique projection cavalier
projection and cabinet p roduction.
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Fig 6.1 3.1 Oblique projection

1. Cavalier projection: Whenever the objects ’ edge parallel to the co -
ordinate axis also will be parallel to the front face that is called as Cavalier
projection in thi s direction of projection is slanted so the points with Z
coordinates are positively project down and to the left on the view plane .
Z co-ordinate negatively projected up and right face . The angle of z axis
can be any of the direction but the distance of the points shifted in a z
direction must be equal to three dimensional view plane z distances that is
called as Cavalier projection .

Fig 6.1 3.2 Cavalier Projection

2. Cabinet projection: In cabinet projection, the lines which are
perpendicular to the projection plane are projected at one and half of the
actual length of the object. The projection is said to be cabinet when an
angle between projection plane and oblique projection is equals to 63.4 .
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Fig 6.13 .3 Cabinet Projection

6.14 ORTHOGRAPHIC PROJECTION
An orthographic projection is a type of parallel projection in which
the projectors are parallel to each other and perpendicular to the plane of
projection. It represents 3 dimensional objects in 2 dimensions. A square
or rectangular viewing volume is form ed for this projection by near, far,
right , left, top and bottom clipping planes. Single orthographic projection
does not provide sufficient information to reconstruct the shape of an
image, therefore, multiple orthographic projection displays a single face of
3D object i.e. viewer can only see the one side of an object at a time.

In multiple projections front surface is parallel to plane of projection.
Projection line of sights is perpendicular to the projection plane. Projectors
are parallel to each other and originate from any point on object.


Fig 6.14 .1 Orthographic Projection

6.15 VANISHING POINTS
A va nishing point is abstract point on the image plane where 2D
projections of a set of parallel lines in 3D space appear to converge or
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intersect i.e. certain set of parallel lines appear to meet at different point
which is known as vanishing point. When the set of parallel lines are
perpendicular to a picture plane then the construction is known as one
point perspective and their vanishi ng point corresponds to the “eye point”
from which the image should be viewed for correct respective geometry.

Types of Vanishing point
1. One point perspective :
As the figure 6.14.1 shows end of the parallel line of hotels or houses
after certain time of period the parallel lines meets each other at some
point that point is called as one perspective vanishing point . Vanishing
points are formed by the intersection parallel lines to one of t he three axis
x, y and z. all three parallel points are perpendicular to the same view
plane at single point.

Fig 6.1 5.1 one point perspective vanishing point

2. Two point perspective :
In two points perspective side of an object vanish or recede towards one of
the two vanishing points on the horizon.

Fig 6.1 5.2 Two point perspective vanishing point
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3. Three point perspective :
In three point perspective, there are three vanishing points where two
vanishin g points are along the horizon and third one is located either
above the horizon or below the horizon.

Fig 6.1 5.3 Three point perspective vanishing point

6.16 VIEW VOLUMES FOR PROJECTIONS
Converting 3D object into 2D is called projection. Projection operation
convert the viewing plane coordinate into 2D coordinate system so that the
object will look like same as it seen in the real world. There are two types
of projection.
1. Parallel projection
2. Perspective projection

Parallel projection :
In this type of projection parallel line of object transforms in parallel lines
only. a parallel projection is also called as relative projection as it
preserves relative plane along with the parallel lines. Acc urate view of
object from multiple sides is view in this projection. This projection will
not preserve realistic representation.

Parallel projection classification: 1 Orthographic Parallel projection 2.
Oblique projection
Orthographic Parallel Projectio n: it is done by projecting points along
parallel lines that are perpendicular to the projection plane.
Oblique projection: oblique projections are obtained by projecting along
with parallel lines that are NOT perpendicular to the projection plane
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Fig 6.16.1 Parallel projection

Perspective projection :
Perspective projection transforms object positions to the view plane while
converging to center point of projection. Perspective projection produces
realistic views but does not preserve relative proporti ons. Projections of
distant objects are smaller than the projections of object of the same size
that are closer to the projection plane.


Fig 6.1 6.2 Perspective projection

View Volume :
A view volume is rectangular area in the view plane which controls
how much of the scene is viewed. T is formed by the window and the type
of projection to be used. View volume is generally defined as a space
between front and backplane.

Only those objects are seen which are within the window volume.
The objects which ar e beyond the view volume it will be not seen. In view
volume the perspective effects depends on the positioning of the center
point of projection. If it is close to the view plane then the object look like
closer and larger but when it is longer to the vie w plane it is look like
smaller a nd far away from the view plane hence the size of the object
affected by the relative position of the object. View volume is bounded by
6 planes left, right, top, bottom, front and back side.




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6.17 SUMMARY
The process of changing size, shape, position and orientation of object is
called transformation. The basic transformation is translation, rotation,
reflection, scaling, shearing. All the transformation which are done in 2D
that are same in 3D with the axi s x, y, z. 3D matrix are represented by 4X
4 matrix with or without homogenous coordinate system. Rotation about
an arbitrary axis can be done through translation and rotation. It can be
done in both directions clockwise and anticlockwise. Reflection throu gh
an arbitrary plane is done with translation, rotation, reflection, inverse
rotation and inverse translation. Study of parallel lines is called as affine
and perspective geometry. Affine transformation can get through
combination of two or more transform ation. Perspective transformation
deals with conventions of 3D world into 2D image. There are three types
of perspective transformations single point, two point and three point
perspective transformation. There are three ways to generating perspective
view s one point, two point and three point perspectives. It is also done in
vanishing point in different perspective projection.

6.18 UNIT END EXERCISE
1. What is 3D transformation? Explain its types.
2. Explain conventions 3D transformation in detail.
3. Explain homogeneous coordinate system in 3D detail.
4. What are multiple 3D transformations ? Explain in detail.
5. How many perspective vanishing points are there ? How it works?
6. Write a short note on r otation about an arbitrary axis .
7. Explain in brief scaling in 3D .
8. Write a short note on cavalier and cabinet projection .
9. Explain reflection through an arbitrary plane .
10. Explain the concept of oblique projection in detail .

6.19 REFERENCES
 Computer Graphics C version 2nd Edition by Donald D. Hearn and
M. Pauline Baker
 Computer Graphics A programming approach 2nd Edition by Steven
Harrington McGraw Hill
 Fundamental of Computer Graphics 3rd Edition by Peter Shirley an
Steven Marschner


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UNIT III
7
VIEWING IN 3D

Viewing in 3D :- Stages in 3D viewing, Canonical View Volume (CVV),
Specifying an Arbitrary 3D View, Examples of 3D Viewing, The
Mathematics of Planar Geometric Projections, Combined transformation
matrices for projections and viewing, Coordinate Systems and matrices,
camera model and viewing pyramid.

Unit Structure
7.0 Objectives
7.1 Stages in 3D viewing
7.2 Canonical View Volume
7.3 Specifying an Arbitrary 3D View
7.4 Examples of 3D Viewing
7.5 The Mathematics of Plana r Geometric Projections
7.6 Combined transformation matrices for projections and viewing
7.7 Coordinate Systems and matrices
7.8 Camera model and viewing pyramid
7.9 Summary
7.10 Unit End Exercises
7.11 List of References

7.0 OBJECTIVES
After going through this chapter, you will be able to:
 define 3D Viewing
 state the Stages in 3D viewing
 specify 3D Arbitrary view
 illustrate the 3D Viewing
 describe the mathematics of planar geometric projections
 derive combined transformation matrices fo r projections and viewing
 classify different types of Coordinate Systems and matrices
 explain Camera model and viewing pyramid

7.1 STAGES IN 3D VIEW ING
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 Modelling transformation
 Viewing transformation
 Projection transformation
 Normalization transformation and Clipping
 View Port transformation


Fig: 7 .1 Stages in 3D Viewing

1. Modelling Transformation:
 Modelling Transformation can be obtained with the help of 3D
transformations.
 It transforms the object (Modelling) coordinates (MC) into 3D world
coordinate system, will generate output as 3D world coordinates
(WC). (3D to 3D).

2. Viewing Transformation:
 The viewing -coordinate system is used in graphics packages as a
reference for speci fying the observer viewing position and the
position of the projection plane .
 It transforms the world coordinate into viewing coordinates (VC) by
performing clipping against canonical view volume.

3. Projection Transformation:
 Projection operation convert s the viewing -coordinate description
(3D) to coordinate positions on the projection plane (2D).
 It projects the coordinates onto projection plane as Projection
Coordinates (PC). (3D to 2D).

4. Normalization Transformation and Clipping:
 The coordinates outside the view volume will get clipped and the
transformation will get normalized and the Normalized Coordinates
(NC) will be generated.


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5. Viewport Transformation:
 It maps the coordinate positions on the projection plane of the output
device.
 It transforms Norma lized Coordinates into viewport is 2D Device
coordinates (DC) or display.

7.2 CANONICAL VIEW VOLUME (CVV)
 Canonical, in computer graphics, is the standard state or behavior of an
attribute. This term is used to refer to perceptions that are distinctive
and/or natural.
 The view volume is instead mapped to a canonical view volume (CVV)
which is a cube that extends from 1 to +1 in each dimension, having
center at the origin. The dimensions of the CVV facilitate a fast and
efficient clipping.

Example of CVV:
CVV is related to P rojection transformation i.e. Orthographic (parallel)
projection and Perspective projection as shown in the below figure:


Figure 7.2: Perspective and Parallel (Orthographic) projection for
CVV demonstrations

Both types of projections, transform a given viewing volume to the
canonical viewing volume (CVV):


Fig: 7 .3
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Note that n and f are typically given as distance which is always positive
and because we are looking towards the –z direction, the actual
coordinates become –n and –f.
Also note the changes in the z -direction. This makes objects further away
from the camera to have larger z values. In other words, CVV is a left -
handed coordinate system.
We need to map the box with corners at (l, b, -n) and (r, t , -f) to the ( -1, -1,
-1) and (1, 1, 1) of CVV.
This is accomplished by the following matrix,
𝑀௢௥௧௛=⎣⎢⎢⎢⎢⎢⎡2
𝑟−𝑙00−𝑟+𝑙
𝑟−𝑙
02
𝑡−𝑏0−𝑡+𝑏
𝑡−𝑏
00−2
𝑓−𝑛−𝑓+𝑛
𝑓−𝑛
000 1⎦⎥⎥⎥⎥⎥⎤

7.3 SPECIFYING AN ARBITRARY 3D VIEW
 We can view the object from the side, or the top, or even from behind.
 Therefore, it is necessary to choose a particular view for a picture by
first defining a view plane.
 A view plane is nothing but the film plane in a camera which is
positioned and oriented for a particular shot of the scene.
 World coordinate positions in the scene are transformed to viewing
coordinates, and then viewing coordinates are projected onto the view
plane.
 The procedure to project from 3D to 2D given a finite view volume,
will be as follows :
 Apply a normalizing transform to get to the canonical view
volume.
 Clip against the canonical view volume
 Project onto the view plane
 Transform into viewport
 To specify an arbitrary view, we should be abl e to place the view
plane anywhere in 3D. Specify the direction of the plane and its
position in the world refe rence coordinate system (WRC).
 A common way to specify an arbitrary view is to specify the
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 A View Reference Point (VRP) which is to point on the plane.
 A View Plane Normal (VPN) which is the normal vector to the
plane.
 A View Up Vector (VUP) which is a vector from which we
determine which way is up.
 Window on View plane :
 PRP: Projection Reference Point
 CW: Center of Window
 COP: Center of Projection (Perspective Projection )
 DOP: Direction of Projection (Parallel Projection )
 PRP and CW are used to determine COP and DOP
 Perspective: COP = PRP
 Parallel: DOP = PRP – CW
 Coordinate Systems :
 WC: World Coordinates - normal, 3 -space (x, y, z)
 VRC: Viewing Reference Co ordinates - defined by VRP, VPN
and VUP



Figure 7 .4: The view plane is defined by VPN and VRP, the v axis is
defined by the projection of VUP along VPN onto the view plane. The u
axis forms the right -handed VRC system with VPN and v.

7.4 EXAMPLES OF 3D VIEWING
 In 3D viewing examples, we are going to set value for VRP, VPN,
VUP, PRP, VRC and F and B of VRC based on the type of
projection along with x, y and z principal axes.
Where,
VRP  View reference point
VPN  View plane normal
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VUP  View up vector
PRP  Projection Reference point
VRC  Viewing reference co -ordinate
F and B of VRC  front and back of VRC

 The notation (WC) or (VRC) is added to the table as a prompt of
the coordinate system in which the viewing parameters is given.
The form of the table is demonstrated for the default viewing
specification used by PHIGS.
 The defaults are shown in below figu re 7.5(a). The view volume
corresponding to these defaults is shown in figure 7.5(b ). If the
type of projection is perspective rather than parallel, then the view
volume is the pyramid as shown in 7.5(c).



Fig: 7 .5 (a) The default viewing specification: VRP is at the origin, VUP is
the y axis, and VPN is the z -axis. This makes the VRC system of u, v and n
coincide with x, y, z world -coordinate system. The window extends from 0
to 1 along u and v, and PRP is at (0.5, 0.5, and 1.0 ). (b) Default parallel
projection view volume. (c) View volume if default projection were
perspective.
 The corresponding values for default view volume is given by,
Viewing parameter Value Comments VRP(WC) (0,0,0) Origin VPN(WC) (0,0,1) Z axis VUP(WC) (0,1,0) Y axis
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PRP(VRC) (0.5,0.5,1.0) Window(VRC) (0,1,0,1) Projection type parallel
 Readers wanting to review how all these parameters interconnect are
encouraged to construct a house, the world coordinate system, and the VRC
system with Tinker Toys, as pictured in Fig. 7.6 The idea is to position the
VRC system in world coordinates as in the viewing example and to
imagine projectors from points on the house intersecting the view plane. In
our experience, this is a valuable way to know 3D viewing concepts.

Fig: 7.6 Stick models useful for understanding 3D Viewing. (a) House
and World -coordinate system. (b) House and VRC system
7.4.1 Perspective Projections
 To obtain the front one -point perspective view of the house shown in Fig.
7.7 we position the center of projection (which can be thought of as the
position of the viewer) at x = 8, y = 6, and = 84. The x value is selected to
be at the horizontal center of the house and the y value to correspond to
the estimated eye level of a viewer standing on the (x, z) plane; the z
value is arbitrary. In this case, z is removed 30 units from the front of the
house (z = 54 plane). The window has been made quite large, to
guarantee that the house fits within the view volume. All other viewing
parameters have their default values, so the overall set of viewing
parameters is as follows:
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 Although the image in Fig: 7. 7 is indeed a perspective projection of the
house, it is very small and is not centered on the view surface. We would
prefer a more centered projection of the house that more nearly spans the
entire view surfa ce, as in Fig: 7 .8 we can produce this effect more easily if
the view plane and the front plane of the house coincide.
 Now, because the front of the house extends from 0 to 16 in both x and y, a
window extending from 1 to 17 in x and y produces reasonable results.
 We place the view plane on the front face of the house by placing the VRP
anywhere in the: = 54 plane; (0, 0, 54), the lower -left front corner of the
house, is fine. For the center of projection to be the same as in Fig: 7. 7, the
PRP, which is in the VRC system, needs to be at (8, 6, 30). Fig: 7.9 shows
this new arrangement of the VRC, VRP and PRP,


Fig: 7. 7 One-point perspective projection of the house & Fig: 7.8 Centered
perspective projection of a house

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Fig: 7.9 The viewing situation for the figure 7.8
which corresponds to the following set of viewing parameters:

 This same result can be obtained in many other ways. For instance, with the
VRP at (8. 6, 54), as in Fig. 7.1 0, the center of projection, given by the PRP,
becomes (0, 0, 30). The window must also be changed, because its definition
is based on the VRC syst em, the origin of which is the VRP. The appropriate
window extends from -9 to 9 in u and from -7 to 11 in v. With respect to the
house, this is the same window as that used in the previous example, but it is
now specified in a different VRC system. Because the view -up direction is
the y axis, the u axis and x axis are parallel, as are the v and y axes. In
summary, the following viewing parameters, shown in Fig. 7.10 , also
produce Fig: 7.8 :
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Fig: 7.10 An alternative viewing situation for fig. 7.8.

 Next, for obtain ing the two -point perspective projection as shown in Fig.
7.11. The center of projection is analogous to the position of a camera
that takes snapshots of world -coordinate objects. With this analogy in
mind, the center of projection in Fig. 7.11 seems to be somewhat above
and to the right of the house, as viewed from t he positive z axis. The
exact center of projection is (36, 25, 74).

Fig: 7.11 Two -point pers pective projection of a house and Fig: 7.12
Isometric projection of a house

 Now, if the corner of the house at (16, 0, 54 ) is chosen as the VRP, then
this center of projection is at (20, 25, 20) relative to it. With the view plane
coincident with the front of the house (the z = 54 plane), a window
ranging from -20 to 20 in u and from -5 to 35 in v is certainly large
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enough t o contain the projection. Hence, we can specify the view of Fig.
7.13 with the viewing parameters:


 This view is similar to, but clearly is no t the same as, that in Fig. 7.11
as is a two -point perspect ive projection, whereas Fig. 7.13 is a one -
point perspective. It is apparent that simply moving the center of
projection is not sufficient to produce Fig. 7.11 . In fact, we need to
reorient the view plane such that it cuts both the x and z axes , by setting
VPN to (1, 0, 1).


Fig: 7.13 Perspective projection of a house from (36, 25, 74) with VPN
parallel to the z axis.



Fig: 7.14 The view plane and VRC system corresponding to
Fig. 7.11





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Thus, the viewing parameters for Fig. 7.11 are as follows:



 Figure 7.11 shows the view plane established with this VPN. There are two
ways to choose a window that completely surrounds the projection, as does
the window in Fig. 7.11 . One can estimate the size of the projection of the
house onto the view plane as shown in Fig. 7.15, to calculate the
intersections of projector s.


Fig: 7.15 Top (plan) view of a house for determining an appropriate
window size


Fig: 7.16 Projection of house produced by rotating VUP.

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with the view plane. A better alternative, however, is to allow the window
bounds to be variables in a program that are determined interactively via a
valuator or locator device. Figure 7.16 is obtained from the same projection
as is Fig. 7.11 but the window has a different orientation.
Fig 7.16 has exactly the same viewing parameters as d oes Fig. 7.11 , except
that VUP has been rotated away from the y axis by about 10°.
 Another way to specify viewing parameters for perspective proje ctions is
suggested in Fig. 7.17. This figure is showed after the way a photographer
might reason about placing a camera.
 Six parameters are needed: the center of projection, which is the camera
position; the center of attention, which is a point at which the camera is meant
(the VPN is the vector from the center of attention to the center of projection);
VUP, the up vector, D, the distance from the center of projection to the
projection plane; W, the width of the window on the projection plane, and H,
the height of t he window on the projection plane.
 The center of attention need not be on the view plane. In this model, the VPN
is always pointed directly at the center of projection, and the view volume is
symmetrical about its center line.


Fig. 7.17 Another way to specify a view, with a camera position
(the center of projection), center of attention, up vector, distance
from the center of projection to the projection plane, and the height
and width of the window on the projection plane. VPN is parallel to
the direction from the center of attention to the camera position.

7.4.2 Parallel Projections :
 We create a front parallel pr ojection of the house ( Fig. 7.18 ) by
making the direction of projection parallel to the z axis and
direction of projection is determined by the PRP and by the center
of the window.
 With the default VRC system and a window of ( -1, 17, -1, 17) , the
center of the window is (8, 8, 0).
 A PRP of (8, 8, 100) offers a direction of projection par allel to the z
axis. Fig. 7.19 shows the viewing situation that creates Fig. 7.18 .

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 The viewing parameters are as follows :




 To create the side view (Fig. 7.19 ), we require th e viewing situation
of Fig. 7.20 , with the (y, z) plane (or any plane parallel to it) as the
view plane. This situation corresponds to the following viewing
parameters:





Fig. 7.18 Front parall el projection of the house.
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Fig. 7.19 Viewing situation that creates Fig. 7.18, a front view of the
house. The PRP could be any point with x = 8 and y = 8.

 The center of the window is at (12, 8, 0 ) in VRC; hence, the PRP has
these same u and v coordinates.
 We create a top view of the house by using the (x, z) plane as the view
plane and VPN as the y axis. The default view -up direction of +y must
be changed; we use the negative x axis . With VRP aga in stated as a
corner of the house, we have th e viewing situation in Fig. 7.22 , defined
by the following viewing parameters:



Fig: 7.20 Parallel projection from the side of the house & Fig: 7.21 The
viewing situation for Fig: 7.20
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Fig: 7.22 The viewing situation for a top view of the house.

 Figure 7.12 is an isometric (parallel orthographic) projection in the
direction ( -1,-1, -1), one of the eight possi ble directions for an isometric.
The following viewing parameters create such an isometric projection:



 A cavalier projection with angle α is specified with



 The window is symmetrical about VRP, which implies that VRP, the origin
of the VRC system, is the window's center. Placing PRP as specified means
that the direction of projection, which is the vector from PRP to the
window center, is (0, 0, 0) - PRP= - PRP, or its negative PRP.
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7.4.3 Finite View Volumes
 In all the example, the view volume has been expected to be infinite.
 The front and back clipping planes, help to determine a finite view
volume.
 These planes, both of which are parallel to the view plane, are at
distances F and B respectively from the view reference point,
measured from VRP along VPN.
 To avoid a negative view volume, we must ensure that F is
algebraically greater than B.

Fig: 7.23 Perspective projection of the house with back clipping plane
at z = 31.
 A front perspective view of the house with the rear wall clipped away
(Fig: 7.23 ) results from the following viewing specification, in which F
and B have been added.
 If a distance is given, then clipping against the corresponding plane is
assumed; otherwise, it is not.
 The viewing specification is as follows :


 If the front and back clipping planes are moved dynamically, the 3D
structure object being viewed can often be discerned more readily than
it can with a static view.





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7.5 THE MATHEMATICS OF PLANAR GEOMETRIC PROJECTIONS
 Planar projections are the subset of 3D graphical projections
constructed by linearly mapping points in three -dimensional projection
plane. The projected point on the plane is chosen such that it is
collinear with the corresponding three -dimensional point and the
centre of projection. The lines connecting these points are commonly
referred to as projectors.
 Type of planar geometric projection is obtained by the proper choice
of a projection plane and a center of projection. The mathematical
properties of ea ch method of projection are related to the properties of
related objects.
 Each type of projection provides variety of visual effects.
 Mathematically, planar projections are linear transformations acting
on a point in three -dimensional space a(x, y, z) to give a point b (u, v)
on the projection plane.
 These transformations consist of various compositions of the five
transformations: orthographic, projection, rotation, shear,
translation and perspective.
 For simplicity, we start by assuming that, in the perspective
projection , the projection plane is normal to the z -axis at z=d, and
that, in the parallel projection, the projection plane is the z=0 plane .
Each of the projection can be defined by a 4 x 4 matrix. Because the
projection matrix can be composed with transformation matrices,
allowing two operations (transform, then project) to be represented as
a single matrix.
 In arbitrary projection planes we derive 4 x 4 matrices for several
projections, beginning with a projection plane at a distance d from the
origin and a point P to be projected onto it.
 To calculate 𝑷𝒑= (𝒙𝒑,𝒚𝒑,𝒛𝒑), the perspective projection of (𝑥,𝑦,𝑧)
onto the projection plane at 𝑧 =𝑑 , we use the similar triangles in
Fig: 7.24 to write the ratios
௫೛
ௗ= ௫
௭; ௬೛
ௗ= ௬
௭ ------------------------ (7.1)

Multiplying each side by d yields

𝑥௣=ௗ.௫
௭=௫
௭/ௗ 𝑦௣=ௗ.௬
௭=௬
௭/ௗ …………….. (7.2)

 The distance 𝒅 is just a scale factor applied to 𝑥௣ and 𝑦௣. The
division by 𝒛 causes the perspective projection of more distant
objects to be smaller than that of closer objects . All values of z are
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projection on the negative 𝑧 axis or between the center of and the
projection plane. The transformation of Eq. (7 .2) can be expressed
as a 4 x 4 matrix:

𝑀௣௘௥=1000
0100
0010
00ଵ
ௗ0 …………………………… (7 .3)


Fig: 7.24 Perspective Projection

 Multiplying the point 𝑃= [𝑥𝑦𝑧1]் by the matrix
𝑀௣௘௥yields the general homogeneous point [𝑋𝑌𝑍𝑊]்:

൦𝑋
𝑌
𝑍
𝑊൪=𝑀௣௘௥.𝑃=൦1000
0100
0010
001/𝑑0൪.቎𝑥
𝑦
𝑧
1቏------------ (7.4)

Or
[𝑋𝑌𝑍𝑊]்= ൣ𝑥𝑦𝑧௭
ௗ൧்------------------ (7.5)
 Now, dividing by W (which is z/d) and dropping the fourth
coordinate to come back to 3D we have

ቀ௑
ௐ,௒
ௐ,௓
ௐቁ=൫𝑥௣,𝑦௣,𝑧௣൯=ቀ௫
௭/ௗ,௬
௭/ௗ,𝑑ቁ; ------------------ (7.6)

 These equations are the correct results of Eq. (7 .1), plus the
transformed z coordinate of d, which is the position of the
projection plane along the z axis.

 An alternative design for the perspective projection places the
projection plane at 𝑧=0 and the center of projection at 𝑧= −𝑑,
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as in fig: 7.25. Similarity of the triangle now gives

௫೛
ௗ=௫
௭ାௗ,௬೛
ௗ=௬
௭ାௗ -------------------------------------------- (7.7)

Multiplying by d, we get

𝑥௣=ௗ.௫
௭ାௗ=௫
௭/ௗାଵ,𝑦௣=ௗ.௬
௭ାௗ=௬
௭/ௗାଵ ------------------------- (7.8)


Fig.7.25 : Alternative Perspective Projection

The matrix is
𝑀′௣௘௥=൦1000
0100
0000
001/𝑑1൪ ------------------------------------------- (7.9)

 This formulation allows d, the distance to the center of projection
to tend to infinity.
 The orthographic projection onto a projection plane at 𝑧=0 is
straightforward. The direction of projection is the same as the
projection -plane normal -the z axis in this case. Thus, point P
projects as
𝑥௣=𝑥,𝑦௣=𝑦,𝑧௣=0 -------------------- (7.10)
 This projection is expressed by the matrix
𝑀௢௥௧ =൦1000
0100
0000
0001൪ ------------------------- (7.11)

 Notice that as d in Eq. (7 .9) tends to infinity, Eq. (7 .9) becomes
Eq. (7 .11).
 𝑀௣௘௥ applies only in the special case in which the center of
projection is at the origin; 𝑀 ௢௥௧ applies only when the direction of
projection is parallel to the z axis.

 In Fig: 7.26 , the projection of the general point 𝑃 = (𝑥,𝑦,𝑧) onto
the projection plane is 𝑃 = ൫𝑥௣,𝑦௣,𝑧௣൯. The projection plane is
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perpendicular to the z axis at a distance 𝑧௣, from the origin, and
the center of projection (COP) is a distance Q from the point
(0,0,𝑧௣). The direction from ( 0,0,𝑧௣) to COP is given by the
normalized direction vector ൫𝑑௫,𝑑௬,𝑑௭൯.𝑃௣is on the line between
COP and P, which can be stated parametrically as

𝑪𝑶𝑷+𝒕(𝑷−𝑪𝑶𝑷), 𝟎≤𝒕≤𝟏 -------------- (7.12)


Fig: 7.26 - The intersection of the line from COP to P=(x,y,z) with
the projection plane at z= 𝒛𝒑 is the projection of the point P. The
COP is distance Q from the point P. The COP is the distance Q
from the point (0, 0, 𝒛𝒑) in direction (𝒅𝒙,𝒅𝒚,𝒅𝒛)

 Rewriting eq. (7.12) as separate equations for the arbitrary point
𝑃’ = (𝑥’,𝑦’,𝑧’) on the line with COP = ( 0,0,𝑧௣) + 𝑄(𝑑௫,𝑑௬,𝑑௭)
yields
𝑥ᇱ=𝑄𝑑௫+(𝑥−𝑄𝑑௫)𝑡, ------------------------- (7.13)
𝑦ᇱ=𝑄𝑑௬+൫𝑦−𝑄𝑑௬൯𝑡, ------------------------- (7.14)
𝑧ᇱ=൫𝑧௣+𝑄𝑑௭൯+(𝑧−(𝑧௣+𝑄𝑑௭))𝑡, ----------------------- (7.15)

 We find the projection 𝑃௣ of the point P, at the intersection of the
line between COP and P with the projection plane, by substituting
𝑧ᇱ=𝑧௣ into Eq. (7 .15) and solving for t;

𝑡= ௭೛ି(௭೛ାொௗ೥)
௭ି(௭೛ାொௗ೥) ------------ --------------------- -------- (7.16)

 Substituting this value of 𝑡 into Eq. (7.13) and Eq. (7 .14) to find
𝑥ᇱ= 𝑥௣ 𝑎𝑛𝑑 𝑦ᇱ=𝑦௣ yields

𝑥௣=௫ି௭೏ೣ
೏೥ା௭೛೏೥೏ೣ
೥೛ష೥
ೂ೏೥ାଵ ------------------------- (7.17)

𝑦௣=௬ି௭೏೤
೏೥ା௭೛೏೥೏೤
೥೛ష೥
ೂ೏೥ାଵ ------------------------------ (7.18)

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 Multiplying the identity 𝑧௣=𝑧௣ on the right -hand side by fraction
whose numerator and denominator are both

௭೛ି௭
ொௗ೥+1 ----------- ----------------------------- (7.19)

 maintains the identity and gives 𝑧௣the same denominator as
𝑥௣and 𝑦௣:

𝑧௣=𝑧௣೥೛ష೥
ೂ೏೥ାଵ
೥೛ష೥
ೂ೏೥ାଵ=ି௭೥೛
ೂ೏೥ା೥೛మశ೥೛ೂ೏೥
ೂ೏೥
೥೛ష೥
ೂ೏೥ାଵ ------------------------------ (7.20)

 Now Eqs. (7.17), (7.18) and (7 .20) can be rewritten as a 4x4
matrix 𝑀௚௘௡௘௥௔௟ arranged so that the last row of
𝑀௚௘௡௘௥௔௟ multiplied by [𝑥𝑦𝑧1]் produces their common
denominator, which is homogeneous coordinate W and is hence
the divisor of X, Y and Z:

𝑀௚௘௡௘௥௔௟=⎣⎢⎢⎢⎢⎢⎡10−ௗೣ
ௗ೥𝑧௣ௗೣ
ௗ೥
01−ௗ೤
ௗ೥𝑧௣ௗ೤
ௗ೥
00−௭೛
ொௗ೥௭೛మ
ொௗ೥+𝑧௣
00−ଵ
ொௗ೥௭೛మ
ொௗ೥+1⎦⎥⎥⎥⎥⎥⎤ --------------------- (7.21)

 𝑀௚௘௡௘௥௔௟ specialized to all three of previously derived projections
matrixes 𝑀௣௘௥,𝑀′௣௘௥and 𝑀௢௥௧ given the following values:



 When 𝑄 ≠∞,𝑀௚௘௡௘௥௔௟ defines a one -point perspective projection. The
vanishing point of a perspective projection is calculated by multiplying the
point at infinity on the z axis , represented in homogenous coordinate as
[0010]் by 𝑀௚௘௡௘௥௔௟.
 Taking this product and di viding by W gives
𝑥=𝑄𝑑௭,𝑦=𝑄𝑑௬,𝑧=𝑧௣
 Given a desired vanishing point (x,y) and known distance Q to the center
of projection these equations uniquely define [𝑑௫𝑑௬𝑑௭], because
ඥ𝑑௫ଶ+𝑑௬ଶ+𝑑௭ଶ=1
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 Similarly it is easy to show that for cavalier and cabinet projection onto
the (x,y) plane with 𝛼 the angle.



 In this section, we have seen how to formulate 𝑀௣௘௥,𝑀′௣௘௥ and 𝑀௢௥௧ , all
of which are special cases of the more general 𝑀௚௘௡௘௥௔௟. In all these
cases, h owever the projection plane is perpendicular to the z axis.

7.6 COMBINED TRANSFORMATION MATRICES FOR PROJECTIONS AND VIEWING
After having all required matrices of projection and viewing, we can now
combine all our transformation into one overall matrix, which will convert
a poi nt from world coordinates to Normalized Device Coordinates (NDC)
while retaining a representation of the relative depth of the points.
MTOT=Awv * Mp * N
Where,
MTOT: - Represents Combined transformation matrices for projection
and viewing
Awv: - Represents Metrics of real -world coordinates
Mp: - Represents Metrics of perspective transformation
N: - Represents Normalization Matrix
This will give us the view volume in its final configuration, called as
Canonical View Volume.

7.7 COORDINATE SYSTEMS A ND MATRICES
A Coordinate system is used to unambiguously represent point, it contains
a reference point (the origin) and three linear independent vectors (the
basis) in the below following figure:

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Fig: 7.27 Basics Vectors

As we have seen in stages in 3D viewing, different stages generate and
work on different coordinate system, given as:
Co-ordinate System:
1. Screen coordinates
2. Local co -ordinate system or 3D modelling coordinate
3. Word coordinate
4. View reference coordinates
5. Normalized projection coordinates
6. 2D device coordinate

Fig: 7 .28 Coordinate systems and how they relate to one another.
The matrices underneath each stage effect the transformation applied
at that stage for the perspective and parallel projections.

 Starting with the coordinate system that is utmost detached from
the real display device, on the left of Fig. 7.28, individual objects
are well-defined in an object -coordinate system, PHIGS
(Programmer's Hierarchical Interactive Graphics System) calls
this as modeling -coordinate system; the term local coordinate
system is also commonly used.
 Objects are transformed into the world -coordinate system, the
system in which a scene or complete object is represented in the
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computer, by the modeling transformation. This system is
sometimes called the problem -coordinate system or application -
coordinate system.
 The view -reference coordinate system is used by PHIGS as a
coordinate system to define a view volume. It is also called the (u,
v, n) system, or the (u, v, VPN) system.
 The Core system used alike , but unidentified , left -handed system.
The left -handed system is used so that, with the eye or camera at
the origin looking toward +z, increasing values of z are beyond
away from the eye, x is to the right, and y is up.
 Other packages, such as Pixar's RenderMan [PIXA88], place
constraints on the view -reference coordinate system, requiring that
the origin be at the center of projection and that the v iew plane
normal be the z axis known as eye-coordinate system, Renderman and
some other systems use the term camera -coordinate system.
 The first three steps of the perspective -projection normalizing
transformation convert from the world -coordinate system into the
eye-coordinate system. The eye coordinate system is sometimes left-
handed.
 From eye coordinates, we next go to the normalized -projection
coordinate sys tem, or 3D screen coordinates, the coordinate system of
the parallel -projection canonical view volume (and of the perspective -
projection canonical view volume after the perspective
transformation).
 The Basic system calls this system 3D normalized device c oordinates.
Sometimes, the system is called 3D logical device coordinates. The
term normalized generally means that all the coordinate values are in
either the interval [0, 1] or 1 -1, 1], whereas the term logical generally
means that coordinate values are in some other prespecified range,
such as [0, 1023], which is naturally defined to resemble some
extensively available device's coordinate system.
 Projecting from 3D into 2D known as 2D device coordinate system,
also called the normalized device -coordinate system, the i mage -
coordinate system or the screen -coordinate system stated by
RenderMan.
 Other terms used include screen coordinates, device coordinates, 2D
device coordinates, and physical device coordinates (in contrast to the
logical device coordinates stated earlier ).
 RenderMan calls the physical fore of the space as raster coordinates.

7.8 CAMERA MODEL AND VIEW ING PYRAMID
 In 3D computer graphics, the view frustum (also called viewing
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on the screen; it is the field of view of the “notional camera”.
 The view frustum is typically obtained by taking a frustum - that is a
truncation with parallel planes - of the pyramid of vision, which is the
adaption of (idealized) cone of vision that a camera or eye would
have to the rectangular viewports typically used in computer
graphics mentioned in below figure: 7.29.
 The exact shape of this region varies depending on what kind camera
lens is being simulated, but typically it is a frustum of a rectangular
pyramid (hence the name). The planes that cut the frustum
perpendicular to the viewing direction are called the near pla ne and
the far plane. Objects closer to the camera than the near plane or
beyond the far plane are not drawn. Sometimes, the far plane is
placed infinitely far away from the camera so all objects within the
frustum are drawn regardless of their distance fr om the camera .


Fig: 7.29 View frustum

 View frustum culling is the process of removing objects that lie
completely outside the viewing frustum from the rendering process.
Rendering these objects would be a waste of time since they are not
directly visible. To make culling fast, it is usually done using bounding
volumes surrounding the objects rather than the objects themselves.

7.9 SUMMARY
 Viewing in 3D is divided mainly into following stages:
 Modelling transformation
 Viewing transformation
 Projection transformation
 Normalization transformation and Clipping
 View Port transformation
 Canonical View Volume (CVV): Canonical, in computer graphics, is
the standard state or Behavior of an attribute. This term is where it is
used to refer to concepts that are unique and/or natural.
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 Specifying an arbitrary view is meant to specify the following:
 A View Reference Point (VRP) which is to point on the plane.
 A View Plane Normal (VPN) which is the normal vector to the
plane.
 A View Up Vector (VUP) which is a vector from which we
determine which way is up.
 Examples of 3D Viewing: In 3D viewing parameters, we set values
for VRP, VPN, VUP, PRP, VRC and F and B of VRC based on the
type of projection along with x, y and z principal axes. Where,
- VRP - View refere nce point
- VPN - View plane normal
- VUP - View up vector
- PRP - Projection Reference point
- VRC - Viewing reference co -ordinate
 Two-point perspective projection of the house
 Two-point perspective projection of the house
 Default perspective projection view volume - pyramid -like
 The Mathematics of Planar Geometric Projections: Planar projections
are the subset of 3D graphical projections constructed by linearly
mapping points in three -dimensional projection plane.
 Mathematically, planar projections are linear transformations
acting on a point in three -dimensional space a(x, y, z) to give a
point b (u, v) on the projection plane.
 Orthographic projection (known as orthogonal projection), is a
means of representing three -dimensional objects in two -
dimensions.
 Com bined transformation matrices for projections and viewing:
MTOT=Awv * Mp * N
 MTOT: - Represents Combined transformation matrices for
projection and viewing
 Awv: - Represents Metric s of real -world coordinates
 Mp: - Represents Metrics of perspective transformation
 N: - Represents Normalization Matrix
 Coordinate Systems and matrices: Different stages generate and
work on different coordinate system, given as:
 Screen coordinates
 Local co -ordinate system or 3D modelling coordinate
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 View reference coordinates
 Normalized projection coordinates
 2D device coordinate
 Camera model and viewing pyramid: the view frustum (also called
viewing frustum) is the region of space in the modeled world that may
appear on the screen; it is the field of view of the “notional camera”.

7.10 UNIT END EXERCISE
Q.1 Explain the concept of viewing in 3D.
Q.2 Explain with neat labelled diagram stages in 3D Viewing Pipeline
with diagram.
Q.3 Explain Canonical View Volume (CVV ) in brief.
Q.4 Explain a procedure which specifies an arbitrary 3D view.
Q.5 Explain the way in which the different examples in 3D viewing is
generated.
Q.6 Briefly explain the mathematics of planar geometry.
Q.7 Explain combined transformation matri ces for projections and
viewing.
Q.8 Explain different c oordinate systems and matrices in detail.
Q.9 Explain the concept of camera model and viewing pyramid with neat
labelled diagram.

7.11 LIST OF REFERENCES
 Computer Graphics - Principles and Practice by J. D. Foley, A. Van
Dam, S. K. Feiner and J. F. Hughes, Publisher - Pearson, 2nd Edition
 Computer Graphics by A. P.Godse and D.A. Godse Technical
Publication.
 Computer graphics with Virtual Reality Systems by Rajesh K. Maurya
 Computer Graphics by Hearn, Baker, Publisher -Pearson, 2nd Edition.
 Mathematical Elements for CG by D. F. Rogers, J. A. Adams,
Publisher - TMH 2ndEdition.




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8

LIGHT: - LIGHT: RADIOMETRY,
TRANSPORT, EQUATION,
PHOTOMETRY

Unit Structure
8.0 Objectives
8.1 What is Light?
8.2 Radiometry
8.2.1 Radiant Energy
8.2.2 Spectral Radiant Energy
8.2.3 Radiant Flux (Radiant power)
8.2.4 Spectral Radiant Flux
8.2.5 Radiant Flux Density (irradiance and radiant Exitance )
8.2.6 Spectral Radiant Flux Density
8.2.7 Radiance
8.2.8 Spectral Radiance
8.3 Transport
8.3.1 Rasterization
8.3.2 Ray Casting
8.3.3 Ray Tracing
8.3.4 Radiosity
8.4 Equation
8.5 Photometry
8.6 Summary
8.7 Unit End Exercise
8.8 List of References

8.0 OBJECTIVES
After going through this chapter, you will be able to:
 define light
 define and understand the concept of Radiometry and its different
variants like Radiant Energy, Spectr al Radiant Energy, Radiant Flux,
Spectral Ra diant Flux, Radiant Flux Density, Spectr al Radiant Flux
Density, Radiance, Spectral Radiance
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modeling techniques like ray tracing , rasterization, ray casting and
radiosity .
 learn and understand the working of ray tracing technique
 state and derive transport equation of light
 describe the concept of Photometry

8.1 WHAT IS LIGHT?
 Light is defined as an electromagnetic radiation. The visible light that
we see is only a tiny fraction of the electromagnetic spectrum,
extending from very low frequency radio waves through microwaves,
infrared, visible and ultraviolet light to x -rays and ultra -energetic
gamma rays.
 Our eyes respond to visible light, identifying rest of the spectrum needs
an arsenal of technical tools ranging from radio receivers to impressive
counters.
 Light is radiant energy. When light is absorbed by a physical object, its
energy is converted into some other form.
 For example: - In microwave oven, when we heats a glass of water
then its microwave radiation is absorbed by the water molecules. The
radiant energy of the microwaves is converted into thermal energy
(heat).
 Similarly, visible light causes an electric current to flow in a
photographic light meter then its radiant energy is transferred to the
electrons as kinetic energy. Radiant energy (denoted as Q) is measured
in joules.

8.2 RADIOMETRY
 Radiometry is the science of measuring radiant energy transfers.
Radiometric quantities have physical meaning and can be directly
measured using proper equipment such as spectral photometers.
 Radiometry is the science of measuring light in any portion of the
electromagnetic spectrum. In practice, the term is typically limited to
the measurement of infrared, visible and ultraviolet light using optical
instruments. There are two aspects of radiometry: theory and practice.
 The practice includes the scientific instruments and materials used in
measuring light, including radiation thermocouples, bolometers,
photodiodes, photosensitive dyes and emulsions, vacuum phototubes,
charged -coupled devices and a plethora of others.

 Units of radiometry:
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general case or in visual units when we are only interested in the fraction
of light in the visible part of the spectrum. Visual units are useful in many
applications such as multimedi a, lightning system etc .

Table below shows radiometric quantities in Radiant and Visual units.
Quantities Symbol Radiant units Visual Units Energy Q Joule (J) Joule(J) Flux F,P() Watt (W) Lumen (lm) Irradiance/Illuminance E W.m-2 Lux Intensity I W.sr-1 Candela (cd) Radiance/Luminance L W.m-2.sr-1 Cd.m-2 Table 8 .1
 Uses of Radiometry :
Following are the application areas based on Radiometry:
 Public: Camera, Photography, TV .
 Biomedical: Optical instrumentation, Medical imaginary.
 Industry: Photovoltaic, Lightning, Security, Non -destructive testing.
 Spatial: Planetary or deep space observation, Satellite design.
 Defense: Identification, Navigation.

8.2.1 Radiant Energy :
• Light is radiant energy. Electromagnetic radiation ( wave and particle,
depends on the measurement) transport energy through spa ce. When
light is absorbed by physical objects, its energy is converted into some
other form. For example , Microwa ve oven heats a glass of water t hen
its microwave radiation is absorbed by the water molecules. The radiant
energy of the microwaves is converted into thermal energy (heat).
• Similarly, visible light causes an electric current to flow in a
photographic light meter when its radiant energy is transferred to the
electrons as kinetic energy.
• Radiant energy is denoted as Q and is measured in joules .

8.2.2 Spectral Radiant Energy :
• A broadband source such as the sun emits electromagnetic radiation
throughout most of the electromagnetic spectrum, from radio waves to
gamma rays. Though , most of its radiant energy is connected within the
visible portion of the spectrum. On the other hand, a single wavelength
laser is a monochromatic source; all of its radiant energy is emitted at
one specific wavelength.
• Spectral radiant energy is the am ount of radiant energy per unit
wavelength interval at wavelength λ. It is given by,
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Defined as, 𝑸𝛌= dQ/dλ …. (8 .1)
• Spectral radiant energy is measured in joules per nanometer .

8.2.3 Radiant Flux (Radiant Power) :
• Energy per unit time is power, which we measure in joules per seconds
or watts.
• Light flows through space, and so radiant power is more commonly
referred to as the time rate of flow of radiant energy, or radiant flux
which is denoted by .
• It is given by, Radiant flux (power) = Radiant energy/time
Defined as,
 = dQ/ dt …..(8 .2)Where,
Q is radiant energy and t is time.

8.2.4 Spectral Radiant Flux :
 Spectral radiant flux is radiant flux per unit wavelength interval at
wavelength λ.
 Spectral radiant flux is denoted by 𝛌, it is given by,
 Spectral Radiant flux = radiant flux/ wavelength
It is measured in watts per nanometer.


8.2.5 Radiant Flux Density (Irradia nce And Radiant Exitance) :
• Radiant flux density is the radiant flux per unit area at a point on a
surface, where the surface can be real or imaginary ( i.e. an exact plane).
• There are two possible conditions:
Irradiance: The flux can be arriving at the surface , in which case the
radiant flux density is referred to as irradiance. The flux can arrive from
any direction above the surface, as indicated by the rays.

Fig: 8 .1 Irradiance
• Irradiance is defined as: 𝑬=𝒅/𝒅𝑨
Where,
 is the radiant flux arriving at the point.
dA is differential area surrounding the point.
Radiant exitance: The flux can be leaving the surface due to emiss ion
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and/or reflection ( fig 8 .2). The radiant flux density is then referred to
as radiant exitance.

Fig: 8 .2 Radiance Exitance

• As with irradiance, the flux can leave in any direction above the
surface.
• The definition of radiant exitance is: M = d /dA
Where,
 Is the radiant flux leaving the point.
dA is the differential area surrounding the point
• Radiant flux density can be measured anywhere in three dimensional is
the space between them (e.g. in air or a vacuum), and inside
transparent media such as water and glass.
• Radiant flux density is measured in watts per square meter.

8.2.6 Spectral Radiant Flux Density :
• Spectral radiant flux density is radiant flux per unit wavelength
interval at wavelength λ.
• When the radiant flux is arriving at the surface, it is called spectral
irradiance denoted by 𝐸λ, and is defined as:
𝑬𝛌=𝒅𝑬/𝒅𝛌 …… (8.3)
• When the radiant flux is leaving the surface, it is called spectral radiant
exitance denoted by 𝑀λ, and is defined as:
𝑴𝛌=𝒅𝑴/𝒅𝛌 ……. (8.4)
• Spectral radiant flux density is measured in watts per square meter
per nanometer.

8.2.7 Radiance:
• Radiance is the radiant flux emitted, reflected, transmitted or received
by a given surface, per unit solid angle per unit projected area.
• Where, a solid angle is the 2D angle in 3D space that an object
subtends at a point. It is a measure of how large the object appears to
an observer looking from that point. In the international system of
units (SI), a s olid angle is expressed in a dimensionless unit called a
steradian .
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• A more formal definition of radiance requires that we think of a ray as
being an extremely small narrow (“elemental”) cone with its apex at a
point on a real or imaginary surface. This con e has a differential solid
angle d  that is measured in steradian.
• We must also note that the ray is intersecting the surface at an angle. If
the area of intersection with the surface has a differential cross -
sectional area dA, the cross -sectional area of the ray is dAcos 𝜃, where
𝜃 is the angle between the ray and the surf ace normal, as shown in
(fig: 8.3) where, the ray cross -sectional area dAcos 𝜃, is called the
projected area of the ray -surface intersection area dA.
• With these preliminaries in mind, we can imagine an elemental cone
d containing a ray of light that is arriving at or leaving a surface ( fig.
8.4 and 8 .5). the definition of radiances is then:
𝑳=𝒅𝟐/[dA(d cos𝜽)] ………(8 .5)


Fig. 8 .3: Ray of light intersecting a Surface

Where,  is the radiant flux,
dA is the differential area surrounding the point,
d is the differential solid angle of the elemental cone,
is the angle between the ray and the surface normal n at the point.
• Unlike radiant flux density, the definition of radiance does not
distinguish between flux arriving at or leaving a surface.

Fig. 8 .4 Radiance (arriving)

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Fig. 8 .5 Radiance (leaving)

• Another way of looking a radiance is to note that the radiant flux
density at a point on a surface due to single ray of light arriving (or
leaving) at an angle  to the surface normal is, d/(dAcos ).
• The radiance at that point for the same angle is then
𝒅𝟐/[dA(d cos𝜽)], or radiant flux density per unit solid angle.
• Radiance is measured inn watts per square meter per steradian.

8.2.8 Spectral Radiance :

• Spectral radiance is radiance per unit wavelength interval at
wavelength λ denoted by𝐿λ. It is defined as:
𝑳𝛌= 𝒅𝟑/ [dA(d cos𝜽)d]…(8.6)
And is measured in watts per square meter per steradian per
nanometer.

8.3 TRANSPORT
 The amount of light transported is measured by flux density,
or luminous flux per unit area on the point of the surface at which it is
measured.
 Light transport theory deals with the mathematics behind calculating
the energy transfers between media that affect visibility.
 Rendering converts a model into an image either by simulating a
method s uch as light transport to get physically based photorealistic
images, or by applying some kind of style as non-photorealistic
rendering .
 The two basic operations in light transport are transport ( how much
light gets from one place to another) and scattering (how surfaces
interact with light).
 Many rendering algorithms have been researched and software used
for rendering are employ ed by using a number of different techniques
to obtai n a final image.
 Tracing every particle of light in a scene is nearly always completely
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impractical and takes more amount of time.
 Even tracing a large enough portion produce an image that takes an
excessive amount of time if the sampling is not intellige ntly limited .
Therefore a few families of more -efficient light transport modeling
techniques have emerged and they are as follows :

8.3.1 Rasterization :
 It assumes an image defined in a vector graphics format (shapes) and
converting it into a raster image (a series of pixels, dots or lines, which,
when displayed together, creates image which was represented via
shapes).

Fig: 8.6 Rasterization
 The rasterized image may then be displayed on a computer display ,
video display or printer , or stored in a bitmap file format.
 Rasterization may refer to either the conversion of models into raster
files, or the conversion of 2D rendering primitiv es such as polygons
or line segments into a rasterized format.

8.3.2 Ray C asting :-
 In 1982 this term was first stated in computer graphics by Scott
Roth to define a technique for rendering constructive solid
geometry models.


Fig: 8.7 Ray casting - Scenario
 It is a technique that alters a limited form of data into a 3D -
projection by tracing the ray from view -point into the viewing
volume.
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 Ray casting is much more simplified way than ray tracing and it
can be done in real time applications.
 Ray casting calculates a color for each pixel i n the image plane by
firing a ray through the view volume.
 It considers the scene as observed from a specific point of view,
calculating the observed image based only on geometry and very
basic optical laws of reflection intensity, and perhaps using Monte
Carlo technique to reduce artifacts.

8.3.3 Ray tracing :
 This is similar to ray casting, but employs more progressive optical
simulation, and uses Monte Carlo techniques to obtain more
realistic results.
 It is a technique for generating an image by tracing the path of light
through pixels in an image plane.
 They apply scan -line rendering method. But there is greater
computational cost involved .
 Ray tracing is mostly used for application where image can be
rendered slowly ahead of time such as in still image, fil m, T.V and
poorly used for application like computer games where speed is
critical.

 How does ray tracing works?



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Fig. 8 .8 Tracing ray from light sources to eye
a) A ray tracing is so named because it tries to stimulate the path that
light rays take as they bounce around within the world.
b) They are outlined through the scene.
c) The objective is to determine the color of each light ray that strikes
the view window before reaching the eye.
d) In this method consider tracing one ray through a scene with one
object as mention in above figure : 8.8.
e) We start from light bulb source then we need to decide, how many
rays to shoot out from the bulb, then for each ray we have to decide in
what direction it is going.
f) There is infinity of directions in which it can travel, to choose one
direction we are tracing a number of photons, some will reach the eye
directly, other will bounce around, some and other will reach the eye
and m any more will never hit the eye at all.
g) For all the rays that will never reach the eye, effort for producing
tracing will be of no use.
h) In order to save our wasted effort, we need to trace only those rays
that are guaranteed to hit the view window and reach the eye.
i) Instead of tracing rays starti ng at light source we trace them backward
starting at the eye.
j) Consider any point on view window whose color we are trying to
determine. Its color depends on the color of light ray s that pass
through that point in view window and reach the eye.
k) For solving this problem we develop two rays : Original ray and
backward ray.
i) If Original ray come directly from light source, then backward
ray will go directly to light source.
ii) If Original ray bounce off the table first, then the backward ray
will also bo unce off the table.
iii) We can clearly see this looking at the above figure again by
reversing the directions of the arrow.
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iv) So backward method does same thing as the original method but
it will not waste any kind of efforts for the rays that will never
reach t he eye.

8.3.4 Radiosity :
 The radiosity method of light interaction was developed by
researchers at Cornell University and Hiroshima University , as a
method for computing radiant heat exchange between surfaces.
 This is a method which attempts to pretend the way in which
directly illumina ted surfaces act as indirect light sources that
illuminate other surfaces. This produces more realistic shading and
seems to better capture the ‘ambience’ of an indoor scene. A
classic example is the way that shows ‘hug’ the corners of rooms.
 It is a globa l illumination algorithm used in 3D computer graphics
rendering.
 It is an application of the finite element method to solve the
rendering equation for scenes with diffused surface.


Fig. 8 .9 Radiosity
 The image on the left was rendered with a typical direct
illumination renderer . There are three types of lighting in this
scene which have been exactly chosen and placed by the artist in an
attempt to create realistic lighting:
o Spot lighting with shadows (placed outside the window to
create the light shining on the floor)
o Ambient lighting (without which any part of the room not
struck directly by a ligh t source would be totally dark)
o Omnidirectional lighting without shadows (to reduce the
flatness of the a mbient lighting).
 The image on the right was reduced using a radiosity algorithm .
There is only one source of light: an image of the sky placed outside
the window. The difference is marked. The room glows with light.
Soft shadows are visible on the floor, and indirect lighting effects are
visible around the room. Besides , the red color from the carpet has
drained onto the grey walls, giving them a somewhat warm advent .
None of these effects were exactly chosen or planned by the artist.
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 Because of this, radiosity is a prime component of leading real -time
rendering methods, and has been used from beginning -to-end to
create a large number of well -known recent feature -length animated
3D-cartoon films .
 There are two types of radiosity:
 Progressive radiosity: - It solves the system iteratively in such a
way that each iteration, we can get an intermediate radiosity value
for the patch.
 Shooting radiosity : - It iteratively solves the radiosity equation by
shooting the light from the patch w ith most error at each step.

8.4 TRANSPORT: EQUATION
The light performance onto an object is analyzed and divided into
three unique components. To actually compute the rays is too expensive to
do in real -time.

Light at a pixel from a light = Ambient + Diffuse + Specular contributor
𝑰𝒍𝒊𝒈𝒉𝒕=𝑰𝒂𝒎𝒃𝒊𝒆𝒏𝒕+𝑰𝒅𝒊𝒇𝒇𝒖𝒔𝒆+𝑰𝒔𝒑𝒆𝒄𝒖𝒍𝒂𝒓
(As given in below diagram )

Fig. 8 .10: Light Model Depending on Various Components

Most important equation for radiance can be obtained as follows:


Fig. 8 .11: Demonstration to calculate total radiance
Total radiance= emitted + reflected radiance
𝑳(𝒙,𝝎𝟎)=𝑳𝒆(𝒙,𝝎𝟎)+න𝒇𝒓(𝝎𝒊,𝒙,𝝎𝟎)𝑳𝒊(𝒙,𝝎𝒊)𝒄𝒐𝒔𝜽𝒊𝒅𝝎𝒊
ା
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In above equation we have, x Surface position 𝜔଴ Outgoing direction 𝜔௜ Incoming illumination direction 𝑓𝑟(𝜔௜𝑥,𝜔௢) BRDF: bidirectional reflectance distribution function 𝐿(𝑥,𝜔଴) Represent total visible surface radiance Table: 8.2

8.5 PHOTOMETRY
 Photometry is the science of measuring visible light in units that are
biased according to the sensitivity of the human eye.
 It is different from radiometry, which is the science of measurement of
radiant energy (including light) in terms of absolute powe r.
 Photometry is a branch of science concerning light in terms of color
apparent by the viewer from the physical inspiration of imposing
photons into the eye and the combined response with the brain.
 The purpose of photometry is to measure light in a way that takes the
sensitivity of human visual system into account. Photometry is essential
for evaluation of light sources and objects used for lighting, signaling,
displays, and other applications where light is envisioned to be seen by
humans.
 It is a quantitative science based on a statistical model of the hu man
visual responses to light that is, our perception of light under wisely
precise conditions.
 The human visual system is an amazingly complex and highly
nonlinear sensor of electromagnetic radiation with wavelength ranging
from 380 to 770 nanometers (nm).
 Light of different wavelengths is perceived as a range of colors ranging
through the visible spectrum 650 nm is red, 540 nm is green , and 450
nm is blue , and so on.
 This task is complicated immensely by the eyes nonlinear response to
light. It varies not only with wavelength but also with the amount of
radiant flux, whether the light is constant or flickering, the spatial
complexity of the scene being perceived, the alteration of the iris and
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Fig. 8 .12: CIE photometric Curve

 The Subjective impression of seeing can be quantified for “normal”
viewing conditions. In 1924, the commission Internatio nale
d’Eclairage (International Commission on Illumination, or CIE)
enquired over one hundred observers to visually match the
“brightne ss” of monochromatic light sources with different
wavelengths under controlled conditions.
 The statistical result so called CIE photometric curve in above figure
shows the photonic luminous effectiveness of the human visual
system as a function of wavelen gth. It provides a weighting function
that can be used to convert radiometric into photometric
measurements.
 Photometric theory does not address how we notice colors. The light
being measured can be monochromatic or a combination or
continuum of wavelength s; the eye’s response is determined by the
CIE weighting function. This emphasizes a vital point: The only
difference between radiometric and photometric theory is in their
units of measurement.

8.6 SUMMARY
 Light is electromagnetic radiation.
• Electromagnetic radiation (which can be considered both wave and a
particle, depending on how you it transports energy through space.
• Radiant energy (denoted as Q) is measured in joules.
 Radiometry is the science of measuring radiant energy transfers. The re
are two aspects of radiometry theory and practice.
• Units of radiometry quantities are expressed in radiant units (such as
watt) in general case or in visual units when we are only interested in
the fraction of light in the visible part of the spectrum.
• Spectral radiant energy is measured in joules per nanometer.
• Radiant Flux (Radiant power): Q is radiant energy and t is time.
• Spectral radiant flux is radiant flux per unit wavelength interval at
wavelength λ.
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• Radiant Flux Density (irradiance and radiant Exitance)
• Spectral radiant flux density is radiant flux per unit wavelength
interval at wavelength λ.
• Spectral radiance is radiance per unit wavelength interval at
wavelength λ denoted by L_λ.
 Light Equation: Light at a pixel from a light = Ambient + Diff use +
Specular contributor.
 A ray tracing is a technique for generating an image by tracing the
path of light through pixels in an image plane.
 A ray casting is a much more simplified version of the ray tracing
and can be easily done in real time.
 Radiosi ty is an application of the finite element method to solve the
rendering equation for scenes with purely diffuse surfaces.
 Photometry is the science of measuring visible light in units that are
weighted according to the sensitivity of the human eye. We se e light
of different wavelengths as a continuum of colors ranging through the
visible spectrum: 650 nm is red, 540 nm is green, and 450 nm is blue,
and so on.

8.7 UNIT END EXERCISES
Q.1 Explain Radiometry in brief.
Q.2 Explain the following terms:
a) Radiant energy
b) Spectral Radiant energy
c) Radiant Flux or Radiant power
d) Spectral Radiant Flux or Radiant power
e) Radiant Flux Density
f) Irradiance radiant exitance
g) Spectral Radiant Flux density
h) Radiance
Q.3 Explain u nits and uses of Radiometry .
Q.4 What is meant by radiance and irradiance ?
Q.5 W rite a short note on rasterization.
Q.6 Explain ray casting technique with diagram in detail.
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Q.8 Write a short note on radiosity.
Q.9 Write a short note on Transport equation.
Q.10 Explain the concept of photometry in detail.

8.8 LIST OF REFERENCES  Computer Graphics by A. P. Godse and D.A. Godse Technical
Publication.
 Computer graphics with Virtual Reality Systems by Rajesh K.
Maurya
 Computer Graphics by Hearn, Baker, Publisher -Pearson, 2nd Edition.
 Computer Graphics - Principles and Practice by J. D. Foley, A. Van
Dam, S. K. Feiner and J. F. Hughes, Publisher - Pearson, 2nd Edition .
 Mathematical Elements for CG by D. F. Rogers, J. A. Adams,
Publisher - TMH 2nd Edition.


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9

COLOR : - COLORIMETRY, COLOR
SPACES, CHROMATIC AD APTATION,
COLOR APPEARANCE

Unit Structure
9.0 Objectives
9.1 What is Color?
9.2 Colorimetry
9.2.1 Colorimetry Concepts in Display Test and Measurement
9.3 Color spaces
9.3.1 RGB Color Model
9.3.2 CMYK Color Model
9.3.3 HSV Color Model
9.3.4 HSL Color Model
9.4 Chromatic Adaptation
9.5 Color Appearance
9.6 Summary
9.7 Unit End Exercises
9.8 List of References

9.0 OBJECTIVES
After studying this Chapter, you will be able to:
• Understand the definition of Color, Colorimetry a nd different
applications of it.
• Learn about Colorimetry Concepts in Display Test and Measurement.
• Know and define the different types of Color spaces, Color Wheel and
Color Model.
• Understand chromatic adaptation and its various forms.
• Learn and state Color Appearance

9.1 WHAT IS COLOR?
 It is defined as an quality of an object or substance with re spects to
light reflected by an object, usually determined visually by
measurement of hue, saturation and brightness of the reflected of
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 It is also defined as an quality of visual insight consisting of any
arrangement of chromatic and achromatic content this attribute can be
described by chromatic color names such as yellow, orange, brown,
red, pink, green, blue, purple, etc., or by achromatic color names such
as white, gray, black, etc., and qualified by bright dim, light, dark,
etc., or by combination of such names.
 Light is a mixture of radiations having different wavelength where
different wavelength implies different colors.
 Light maybe decomposed into a spectrum that is the separation of
photos (light quanta) with respects to the different wavelength i.e.
different colors. Illustrated in fig : 9.1
 Spectrum is measured as Watt per square meter per nanometer. The
spectrum is condition that has infinite number s of independent
elementary colors present in it.


Figure 9 .1 Spectrum of Light


9.2 COLOR: COLORIMETRY
 Science related to the perception of colors is known as Colorimetry .
 The Science of Colorimetry is used to quantify the response of the
human visual system and match human color perception for
applications in a variety of industries.
o Display Manufacturing : - Quality control for industrial
production lines and incoming inspection of display glass.
Display calibration for LED, LC D, Plasma, projection, CRT
displays.
o Broadcasting: - Measuring and standardizing video walls for
color accuracy, uniformity of brightness and white balance.
o Graphic Design and Computer Animation: - Professionals
who rely on color accuracy and precision co lor measurement
benefit from understanding Colorimetry.




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9.2.1 COLORIMETRY CONCEPTS IN DISPLAY TEST AND
MEASUREMENT:

 Color: Color perception is naturally independent . The objective
capacities of the color of a source object i s alleged by a standard human
observer that also can be measured . Various standard human observers
are defined in the disciple of Colorimetry. According to standard
model, the perceived color of a given spot can be reduced to a three -
dimensional value. The three di mensions of color can be described as
color of brightness, hue and purity or saturation.
 Chromaticity: A two -dimensional explanation of color has the mixture
of hue and purity, neglecting the third dimension of brightness. The
luminance (brightness) and chromaticity of a spot on a display, taken
together, provide a complete depiction of its color.
 Gamut mapping (RGB plotting): All colors that are produced by a
display device are created by combination of Red, Green and Blue
colors . Each display color can be described in terms of the amount of
R, G and B primaries present. If the chromaticity directs the three
primaries are plotted in a chromaticity diagram, the triangle enclosed
by these po ints represents the full range of colors reproducible by the
display. This range is the displays of color gamut.
 Correlated Color Temperature (CCT): This is a metric used to
define the color of a white light by comparing its chromaticity to that of
an idealized radiant source, known as a black body. The color of a
bright source (which glows due to heat) depends upon its temperature;
lower temperature sources are bluer . The CCT of a white light is the
high temperature of the black body which almost closely matches its
chromaticity.
 White Balance, Gray Balance: One of the vital features of color
depiction is the position of the white point. The chromaticity of a
specified white light, to which the observer is presumed to be
improved . Light sources with different CCT have slightly different
chromaticity’s , but once the location of the desired white point is
specified, the proportion of R, G a nd B primaries required to reproduce
are classically non-linear, a different gray balance may be required to
reproduce the same chromaticity at a lower luminance level.
 Dominant Wavelength and Purity: These values taken together to
represent a substitute description of chromaticity. Dominant
wavelength corresponds to hue, while purity corresponds to saturation.
The relationship between wavelength and hue can be tacit in terms of
the colors of the visible spectrum, as perceived in the rainbow: Shorter
wavele ngths resemble to violet and blue hues; medium wavelengths to
greens and yellow; longer wavelengths to orange and red hues.



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9.3 COLOR SPACES
 A range of colors that can be created by the primary colors of pigment
and theses colors then define a specific color space.
 It is a way to represent colors, usually used in relation to computers or
graphics boards. Color includes levels of grey. Number of bits specifies
color space. 1 bit color is black and white combination. Grey scale
pattern refers 2 bits to generate black, dark grey, light grey, white
colors.
 Color space, also known as the color model (or color system), is an
abstract mathematical model which simply descr ibes the range of
colors as tuples of numbers, typically as 3 or 4 values or color
components (e.g. RGB). It is an elaboration of the co -ordinate system
and sub -space. Each color in the system is represented by a single dot.
 A Color wheel is a tool that of fers a visual representation and
relationships between all possible hues.
 The primary colors are arranged around a circle at equal (120 degree)
intervals. Color Wheels often depict “Painter’s Colors” primary colors,
which leads to a different set of hues.

Fig 9 .2: Color Wheel

 A Color model is an abstract mathematical model describing the
way colors can be represented as tuples of numbers, typically as
three or four values or color components. When this model is
associated with a precise description of how the components are to
be interprete d (viewing conditions, etc.), the resulting set of colors
is called color space. This section describes ways in which human
color vision can be modeled.
 There are three different color models that describes the different
perceived characteristics of color are RGB model, CMY model,
HSV or HLS Model.


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9.3.1 RGB Color model :
 The RGB color model is an additive color model in which red, green,
and blue light are added together in various ways to reproduce a broad
array of colors. The name of the model comes from the initials of the
three additive primary colors, red, green, and blue .
 The main purpose of the RGB color model is for the sensing,
representation, and display of images in electronic systems, such as
televisions and computers, though it has also been use d in a
conventional photography. Before the electronic age, the RGB color
model already had a solid theory behind it, based on the human
perception of colors .



Fig 9 .3: RGB color model

 In the RGB color model, we use red, green, and blue as the three
primary colors. We do not actually specify what wavelengths these
primary colors correspond to, so this will be different for the different
types of output media, for example, different monit ors, films,
videotapes, and slides.
 This is an additive model since the phosphors are emitting light. A
subtractive model would be the one in which the color is the reflected
light. We can represent the RGB model by using a unit cube. Each
point in the cube (or the vector where the other point is the origin)
represents a specific color. This model is the best for setting the
electron guns for a CRT ( refer Figure : 9.3)

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9.3.2 CMY Color model :
 CRTs produce color by the emission and use of the RGB model.
Printers produce color by the reflective light, so it is a subtractive
process and uses a model based on the colors i.e. cyan, magenta, and
yellow (CMY).
 CMYK (short for cyan, magenta, yellow an d key or black), or the
process color (the four -color model) is a subtractive color model used
in the color printing, also used to descri be the printing process itself.
 The CMYK model works by partially or entirely masking, certain
colors on the typically white background (that is absorbing particular
wavelengths of light). Such a model is called subtractive because the
inks subtract the brightnes s from white ( see Figure 9.4)


Fig 9 .4: CMY color model

 Remember that cyan -green+ blue, go the light reflected from a cyan
pigment has no red component, that is, red is absorbed by cyan.
Similarly, magenta subtracts green, and yellow subtracts blue.
 Printers usually use four colors: cyan, yellow, magenta, and black. This
is because cyan, yellow, and magenta together produce a dark grey
rather than a true black. The conversion between the RGB and CMY is
easily computed as
 Cyan 1 -Red
Magenta -1-Green
Yellow 1 -Blue
 The reverse mapping is similarly, obvious.
Red-1-Cyan
Green -1-Magenta
Blue -1-Yellow
9.3.3 HSL C olor model:
 An HSL model is based on the parameters hue, saturation, and
lightness .
 The HSL is another way to describe a color with the three parameters.
The RGB model is the way computer screens a work , but not very
intuitive.
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 The nicest application of this color model is that you can easily create
the rainbow gradients or change the color, lightness or saturation of an
image with this color model.



Fig 9 .5: Hue in HSL C olor model

 The hue indicates the color sensation of light. In other words, if the
color is red, yellow, green, cyan, blue, and magenta, this representation
looks almost the same as the visible spectrum of light, except on the
right is now the color magenta (the combi nation of red and blue),
instead of violet (light with a frequency higher than blue).
 The hue works circular, so it can be represented on a circle instead. A
hue of 360° looks the same again as a hue of 0 °. The saturation
indicates the degree to which the hue differs from a neutral gray.
 The values run from 0 percent (which is no color) to 100 percent
(which is the fullest saturation) of a given hue at a given percentage of
the illumination. The more the spectrum of the light is concentrated
around one wa velength, the more saturated the color will be
 The lightness indicates the illumination of the color. At 0 percent, the
color is completely black, at 50 percent, the color is pure, and at 100
percent, it becomes white ( see Figure 9.5).
 The HSL, color mode l is, for example , used in Corel Paint Shop Pro*
Photo X2 color picker and other color -processing software packages.

9.3.4 HSV Color model :
 The HSV (hue, saturation, value), also called the HSB (hue, saturation,
brightness) model, defines a color space in terms of three constituent
components: hue, saturation, and value .
 The HSV color model is more intuitive than the RGB color model.
 The user specifies a color (hue) and then adds white or black. Changing
the saturation parameter corresponds to adding or subtracting white and
changing the value parameter corresponds to adding or subtracting
black.
 A 3D representation of the HSV model is derived from the RGB mode
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cube. If we look at the RGB, the cube along the gray diagonal, we can
see a hexagon that is the HSV hexcone. The hue is given by an angle
about the vertical axis with red at 0°, yellow at 60°, green at 120", cyan
at 180°, blue at 240°, and magenta at 300%. Note that the
comp lementary colors are 180° apart.


(a)


(b)
Fig 9 .6: 3D presentation of an HSV model
(a) Color model (b) C olor cone

 Saturation S, varies between 0 and 1, is the ratio of purity of a related
hue to its maximum purity at saturation equal to 1. Also S=0 yields the
grayscale corresponding to value V of HSV that is the diagonal of the
RGB cube with V=0 being black and V=1 corresponding to white.
 The HSV color whee l (refer to Figure 9 .6(a)) is used to pick the
desired color. The hue is represented by a circle in the wheel. A
separate triangle is used to represent the saturation and the value. The
horizontal axis of the triangle indicates the value, and the vertical axis
represents the satura tion. When you need a particular color for your
picture, first you need to pick a color from the hue (the circular region),
and then from the vertical angle of the triangle, you can select the
desired saturation. For brightness, you can select the desired value from
the horizontal angle of the triangle.
 Sometimes, the HSV model is illustrated as a cylindrical or conical
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object (refer to Figure 9 .6(b)). When it is represented as a conical
object, a hue is represented by a circular part of a cone. The cone is
usually represented in a 3D form. A saturation is calculated using the
radius of the cone, and the value is the height of the cone. A hexagonal
cone can also be used to represent an HSV model. The advantage of the
conical model is that it is able to repre sent the HSV color space in a
single object. Due to the 2D nature of computer interfaces, the conical
model of HSV is best suited for selecting colors for computer graphics.
 The colors used in HSV can be clearly defined by the human perception
that is not always the case with RGB or CMYK.
 The HSV color space is widely used to generate high -quality computer
graphics. In simple terms, it is used to select different colors needed for
a particular picture. An HSV color wheel is used to select the desired
color.
 A user can select a particular color needed for a picture from the color
wheel. It gives the color according to the human perception.

9.4 CHROMATIC ADAPTATION
 The surroundings in which viewer objects and images has a larger
effect on how we perceive those objects/ the range of viewing
environment (i.e., by mean of light) is very large, from sunlight too
moonlight or from candle light to luminous light. The lightning
condition is not only involving a very high range of light but als o varies
greatly in the range of color which emits light.
 A human visual system accommodates these change in the environment
through a process called as adaptation.
 There are three types of adaptation:
- Light adaptation
- Dark adaptation
- Chromatic adaptation
 Light Adaptation:
1) It refers to the change occurs when we move from very dark to very
light environment i.e., dark -> light
2) When this happens we are dazzled at first by the light but soon we
adapt to the new situation and then we begin to distinguish objects in
our environment.
 Dark Adaptation:
1) It refers to the change occurs when we move from very light to very
dark enviro nment i.e., light -> dark
2) When this happens we see very little at first but after some time the
details of an objects starts appearing in front of us.
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that of light adaptation.
 Chromatic Adaptation:
1) It refers to the human’s ability to adjust and largely ignore differences
in the color of the illumination. Although, we are able to largely ignore
the changes in the viewing environment but we are unable to do it
completely. For example, color appears much more colorful in a sunny
day as compare to a cloudy day.
2) Chromatic adaptation is the ability of the human visual system to
discount the color of a light source and to approximately preserve the
appearance of an object. For e xample, a white piece of paper appears to
be white when viewed under sky light and tungsten light (light under a
light bulb). However, the measured tri -stimulus values are quite
different for the two viewing conditions for e.g., sky light is bluer and
contains shorter wavelength energy than tungsten light w hich is
illustrated in fig: 9.7.

Fig 9 .7: Chromatic adaptation under daylight and blue sky
light
3) Chromatic adaptation is the biological equivalent of a white balancing
operation that is available on most of the modern cameras. It allows
white objects to appear white for a large number of lightning
conditions.
4) Digital imaging systems, such as digital cameras and scanners, do not
have the ability to adapt to the light source. Scanner usually use
fluorescent light sources. For digital cameras the light source varies
with the scene, and sometimes within a scene. Therefore, to achieve the
same appe arance of the original or original scene under different
display conditions (such as a computer monitor or a light booth), the
captured image tri -stimulus values have to be transformed to take into
account the light source of the display viewing conditions such
transformation are called chromatic adaptation transforms (CATs)
5) The adapting illumination can be measured off a white surface of a
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scene. In a digital image, the adapting illumination came be
approximated as the maximum t ri-stimulus values of the scene.

9.5 COLOR APPEARANCE
 Colorimetry allows us to precisely require and communicate color in
device independent manner and chromatic adaptation allows us to
predict color matches across changes in illumination but these tools are
still inadequate to define how color actually look like.
 A Color appearance Model provides mathematical formulae to
transform Physical measurements of the stimulus and viewing
environment into Correlates of perceptual attributes of color (eg.
lightness, Chroma, hue, etc.) . Based on this color s are divide in four
types, viz. Nominal, Ordinal, Interval and Ratio . Below Fig 9.8
Shows this four types along with examples. This types are based on the
values found after plotting as shown in the below fig: 9.9.


Fig 9 .8: Stimulus Perceptual Attributes of Color


Fig 9 .9: Plotting used for stimulus perceptual attributes of color

 To predict the actual perception of an object , we required to know more
information about an environment. The human visual system is
constantly adapting to t his changing environment thus the perception of
color is highly affected by such changes.
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 There are different parameters used for color appearance which are
given as follows:
- HUE
- Brightness
- Lightness
- Colorfulness
- Chroma
- Saturation
 HUE: It is an Attribute of a visual sensation according to which an area
appears to be similar to one of the perceived colors i.e. pure color: red,
yellow, green, and blue, or to a combination of two of them. Hue is a
more technical definition of our color per ception which can be used to
communicate color ideas.
 Brightness: It is an Attribute of a visual sensation according to which
an area appears to emit more or less light. It is referred to as the
absolute level of the perception.
 Lightness: It is a represen tation of variation in the perception of a color
or color space's brightness. It is referred to as relative brightness
normalized for changes in the illumination and viewing conditions.
Lightness defines a range from dark (0%) to fully illuminate (100%).
Any original hue has the average lightness level of 50%. Lightness is
the range from fully shaded to fully tinted. We can lighten or darken a
color by changing its lightness value
 Colorfulness: It is an a ttribute of a visual sensation according to which
the perceived color of an area seems to be more or less chromatic (e.
multiple color variations).
 Chroma: Chroma is a component of a color model. There's a blue
yellow and a red -green chroma component.
 Saturation: Saturation is used to determine certain color and measured
as percentage value. Saturation defines a range from pure color (100%)
to gray (%) at a constant lightness level.
 A pure color is fully saturated. A de -saturated image is said to be dull,
less colorful or washed out but can also make these at tributes are also
unable to predict the changes in impression of being softer.
A color appearance model (abbreviated CAM) is a mathematical model
that seeks to describe the perceptual aspects of human color vision, i.e.,
viewing conditions under which the appearance of a color does not tally
with the corresponding physical measurement of the stimulus source.
(In contrast, a color model defines a coordinate space to describe
colors, such as the RGB and CMYK color models.)
Thus there are several mathematica l models are used to improve the
color appearance.
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Few color appearance models are listed below:
1. CIELAB Model - first color appearance model to focus on overall
color improvement of an object
2. Nayatani et al Model - focuses on illumination engineering and the
color rendering properties of light sources.
3. Hunt Model -focuses on color image reproduction
4. RLAB Model -focuses on color image reproduction
5. LLAB Model -focuses on color image reproduction
6. CIECAM97A Model -focuses on color image reproduction
7. Model -focuses on hue color formulation and suited for gamut
mapping
8. ICtCp Model -improves the original IPT by exploring higher dynamic
range and larger color gamut.

9.6 SUMMARY
 A color model is an abstract mathematical model that describes the
way colors can be represented. It can be represented as tuples of
numbers, typically as the three or four values of the color components.
A CIE chromaticity diagram is an international standard for the
primary colors established in 1931.
 Colorimetry is a science of color measurement and color appearance.
 Colorimetry applications :
 Display Manufacturing
 Broadcasting
 Graphic Design and Computer Animation
 Color space , also known as the color model (or color system), is an
abstract mathematical model which simply describes the range of
colors as tuples of numbers, typically as 3 or 4 values or color
components (e.g. RGB).
 Chromatic adaptation is the ability of the human visual system to
discount the colo r of a light source and to approximately preserve the
appearance of an object. There are three types of adaptation:
 Light adaptation
 Dark adaptation
 Chromatic adaptation
 A Color wheel is a tool that provides a visual representation of the
relationships bet ween all possible hues.
 A Color appearance Model provides mathematical formulae to
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environment into Correlates of perceptual attributes of color (eg.
lightness, Chroma, hue, etc.)

9.7 UNIT END EX ERCISES
Q.1 What is color?
Q.2 Explain Colorimetry and its different applications in brief.
Q.3 Explain Colorimetry Concepts in Display Test and Measurement.
Q.4 What is meant by Colorspaces and Color Wheel?
Q.5 Explain following color model with diagram:
a) RGB
b) CMY
c) HSV
d) HSL
Q.6 Explain chromatic adaption in brief.
Q.7 Explain Color appearance in brief.

9.8 LIST OF REFERENCES
 Computer Graphics by A. P.Godse and D.A. Godse Technical
Publication.
 Computer graphics with Virtual Reality Systems by Rajesh K. Maurya .
 Computer Graphics by Hearn, Baker, Publisher -Pearson, 2nd Edition.
 Computer Graphics - Principles and Practice by J. D. Foley, A. Van
Dam, S. K. Feiner and J. F. Hughes, Publisher - Pearson , 2nd Edition
 Principles of Interactive Computer Graphics by William M. Newman
and Robert F. Sproull, Publisher - TMH, 2nd Edition.


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UNIT IV
10

VISIBLE -SURFACE DETERMINATION
AND PLANE CURVES AND SURFACES

Unit Structure
10.0 Objectives
10.1 Techniques for efficient Visible -Surface Algorithms,
10.2 Categories of algorithms
10.2.1 Back face removal
10.2.2 Scan -line method
10.2.3 The z -Buffer Algorithm
10.2.4 Area sub-division method (Warnock’s Algorithm)
10.2.5 BSP trees
10.2.6 Painter’s algorithms (depth sorting)
10.2.7 Visible -Surface Ray Tracing
10.3 Comparison of the methods
10.4 Summary
10.5 Questions
10.6 References
References

10.0 OBJECTIVES
1. To know the meaning of Visible -surface detection
2. To distinguish among image -space and object -space approach for
visible -surface determination.
3. To develop and understand the depth -buffer method for visible -
surface determination.
4. To develop and explain the Scan -line method for visible -surface
determination.
5. To understand the Area -Subdivision method for visible -surface
determination.




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10.1 VISIBLE -SURFACE DETERMINATION
10.1.1 Techniques for efficient Visible -surface Algorithms
The techniques to perform visible surface algorithms efficiently are
discussed in the following sections.
1. Coherence
The coherence is defined as the degree to which parts of an environment,
or its projection exhibit local similarities. Such as similarities in depth,
colour, texture and so on. To make algorithms more efficient we can
exploit these similarities when we reuse calculations made for one part of
the environment or a picture for ot her nearby parts, either without changes
or with some incremental changes. Different kinds of coherence we can
use in visible surface algorithms are.
i. Object coherence : if one object entirely separates from another,
comparisons may need to be done only betw een the two objects, and
not between their components faces or edges.
ii. Face coherence: Usually surface properties vary smoothly across a
face. This allows the computations for one part of face to be used with
incremental changes to the other parts of the fa ce.
iii. Edge coherence: The visibility of edge may change only when it
crosses a visible edge or penetrates a visible face.
iv. Implied edge coherence: If one planar face penetrates another their
line of intersection can be determined from two points of intersecti on.
v. Area coherence: A group of adjacent pixels is often belonging to the
same visible face.
vi. Span coherence: It refers to a visibility of face over a span of adjacent
pixels on a scan line. It is special case area coherence.
vii. Scan line coherence: The set of visible object spans determined for
one scan line of an image typically changes very little from the set on
the previous line.
viii. Depth coherence: Adjacent parts of the same surface are typically
same or very close depth. Therefore, once the depth at one poi nt of the
surface is determined, the depth of the points on the rest of the surface
can be determined by at the simplest incremental calculation.
ix. Frame coherence: Pictures of the same scene at two successive
points in time are likely to be quite similar, e xcept small changes in
object and view ports. Therefore, the calculations made for one
picture can be reused for the next picture in as sequence .
2. Perspective Transformation :
Visible -surface determination is done in a 3D space prior to the projection
into 2d that destroys the depth information needed for depth comparisons,
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Due to this projector are parallel to the Z axis in parallel projections or
emanate from the origin in persp ective projections.
In parallel projection, when x1=x2 and y1=y2 we can say that points are
on the same projector.
However, in perspective projection we must perform four divisions:
x1/z1=x2/z2 and y1/z1=y2/z2 to determine whether the points are on the
same projector.
These divisions can be avoided by first transforming a 3D object into the
3D screen -coordinate systems, so that the parallel projection of the
transformed object is the same as the perspective projection of the
untransformed object.
3. Extents and bounding Volumes :
a. Two objects with their projections and the rectangular screen extents
surrounding the projections.
b. It is easier to check the overlapping of extents than the projectio ns,
and we can say that when the extents are not overlapping then
projections are also not overlapping. Therefore, extents must be
compared first and then the projections must be compared only if the
extents are overlapped. This avoids unnecessary comparisons in
checking the overlapping of projections.
c. Extents used to surr ound the object themselves rather than their
projections; in this case, the extent become solid and are commonly
known as bounding volumes. In this case extent can be used to check
whether two objects are overlapped or not.
d. Extent and bounding volumes are used to determine whether project or
intersects an object or not.
4. Back -Face Culling
a. When an object is approximated by a solid polyhedron, its polygonal
faces completely enclose its volume. In such case, if none of the
polyhedron’s interior is exposed by the front clipping plane, then
those polygons whose surface normal point away from the observer
lie on a part of the polyhedron whose visibility is completely blocked
by other closer polygons.
b. Polygons (A, B, D, F) in gray are eliminated, whereas front -facing
polygon ( C, E,G,H) are retained.
c. These invisible back facing polygons can be eliminated from further
processing. Such a technique to eliminate the back facing polygon
from further processing is known as Back -face Culling.

5. Spatial Parti tioning :
a. In this technique, subdivision rule is applied to break down a large
problem i nto a number of smaller ones. T his object and their
projections are assigned to spatially coherent groups as a pre -
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which object intersect with a projector. Because now it is necessary to
test only the objects lying within the partitions which intersect with
projector.
b. When objects are unequally distributed in space the adaptive
partitioning is more suitable. B ecause it allows variable size of each
partition.
6. Hierarchy
a. In hierarchical structure different levels are assigned to the object and
there is a parent -child relationship between the objects.
b. In this structure, each child is considered as a part of its par ent. This
allows to restrict the number of object comparison needed by a
visible -surface algorithm.
c. If the parent level object is fail to intersect, the lower level objects
belongs to the parent do not need to be tested for intersection.
10.1.2. Object Spa ce Method and Image Space Method : Object Space method Image Space method 1. It deals with object definition directly 1. It is a pixel-based method. It is concerned with the final image, what is visible within each raster pixel. 2. Here surface visibility is determined. 2. Here line visibility or point visibility is determined. 3. It is performed at the precision with which each object is defined, No resolution is considered. 3. It is performed using the resolution of the display device. 4. Calculations are not based on the resolution of the display so change of object can be easily adjusted. 4. Calculations are resolution base, so the change is difficult to adjust. 5. These were developed for vector graphics system. 5. These are developed for raster devices. 6. Object-based algorithms operate on continuous object data. 6. These operate on object data. 7. Vector display used for object method has large address space. 7. Raster systems used for image space methods have limited address space. 8. Object precision is used for application where speed is required. 8. There are suitable for application where accuracy is required. 9. It requires a lot of calculations if the image is to enlarge. 9. Image can be enlarged without losing accuracy. 10. If the number of objects in the scene increases computation time also increases. 10. In this method complexity increase with the complexity of visible parts. munotes.in

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10.2 CATEGORIES OF ALGORITHMS
10.2.1 Back - face removal :
1. A polygon has two surfaces, a front and a back just as a piece of
paper does. We might picture our polygons with one side painted
light and the other painted dark. But the question is “how to find
which surface is light or dark”.
2. When we are look ing at the light surface, the polygon will appear to
be drawn with counter clockwise pen motions, and when we are
looking at the dark surface the polygon will appear to be drawn with
clockwise pen motions.
3. Let us assume that all solid objects are to be constructed out of
polygons in such a way that only the light surfaces are open to the
air; the dark faces meet the material inside the object. This means
that when we look at an object face from the outside, it will appear
to be drawn counter clockwise .
4. If a polygon is visible, the light surface should face towards us and
the dark surface should face away from us. Therefore, if the
direction of the light face is pointing towards the viewer, the face is
visible, otherwise the face is hidden and should be r emoved.
5. The direction of the light face can be identified by examining the
result
N.V > 0
Where N: Normal vector to the polygon surface with cartesian
components(A,B,C).
1V: A vector in the viewing direction from the eye position.
6. We know that, the dot pro duct of two vector gives the product of the
length of the two vectors times the cosine of the angle between
them.
7. This cosine factor is important to us because if the vectors are in the
same direction (0≤Θ<π/2), then the cosine is positive, and the overall
dot product is positive. However, if the direction is opposite
(π/2<Θ≤π), then the cosine and the overall dot product is negative.
8. If the dot product is positive, we can say that the polygon faces
towards the viewer ; otherwise it faces away and should be removed.
9. In case, if object description has been converted to projection
coordinates and our viewing direction is parallel to the viewing zv
axis, then V=(0, 0, Vz) and V.N = Vz C
10. So that we only have to consider the sign of C, the Z component of
the norm al vector N. Now, if the z component is positive, then the
polygon faces towards the viewer, if negative, it faces away.


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10.2.2 Scan Line Algorithm :
A scan line method of hidden surface removal is another approach of
image space method. It is an extension of the scan line algorithm for
filling polygon interiors. Here, the algorithm deals with more than one
surfaces. As each scan line is processed, it e xamines all polygon surfaces
intersecting that line to determine which are visible. It does the depth
calculation and finds which polygon is nearest to the view plane. Finally,
it enters the intensity value of the nearest value of the nearest polygon at
that position into the frame buffer.

1. We know that scan line algorithm maintains the active edge list. This
active edge list contains only edges that cross the current scan line,
sorted in order of increasing x. The scan to indicate whether a position
along a scan line is inside or outside of the surface. Scan lines are
processed from left to right. At the leftmost boundary of a surface, the
surface flag is turned ON, and at the rightmost boundary, it is turned
OFF.

Figure 1: Example 1 Scan Line Algorithm
2. Example 1 : The figure illustrates the scan line method for hidden
surface removal. The active edge list for scan line 1 contains the
information for edges AD, BC, EH an FG. For the positions along this
scan line between edges AD and BC, only the flag for s urface S 1 is
ON. Therefore, no depth calculations are necessary, and intensity
information for surface S 1 is entered the frame buffer. Similarly,
between edges EH and FG, only the flag for s urface S 2 is ON and
during that portion of scan line intensity inf ormation for surface S 2 is
entered the frame buffer.
3. For scan line , the active edge list contains edges AD, EH, BC, FG.
Along the scan line 2 from edge AD to edge EH, only the flag for
surface S 1 is ON. However, between edges EH and BC, the flags for
both surface are ON. In this portion of scan, line 2, the depth
calculations are necessary. Here we have assumed that the depth of S 1
is less than the depth of S 2 and hence the intensities of surface S 1 are
loaded into the frame buffer. Then for edge BC to edge FG portion of
scan line 2 intensities of surface S 2 are entered the frame buffer
because buffer during that portion only flag for S 2 is ON.
4. This algorithm is based on the Image -space method and concept of
coherence. As its name suggests itself Scan -line algorithm, so it
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processes one line at a time rather than processing one pixel(a point
on raster display) at a time.
5. Edge list table : This list maintains the record of all the edges by
storing their endpoint coordinates. The x -coordinate that we choose,
whose Y -coordinate = Ymin.

Figure 2: Edge list table

6. Active edges table(list): This table contains all those edges of the
polygon tha t are intersected(crossed) by the current scan -line. The
edges are dropped into the table in a sorted manner(Increasing value
of x).
7. Polygon table(list) : This list consists of:
a. Polygon Id.
b. Plane equation.
c. Color Information of the surface.
d. Flag of surface(on/off)].
8. Example 2 : Here, two overlapped polygons are given which are
intersected by three Scan -lines S 1, S 2, S 3 respectively. So, What
happens if the Scan -line algorithm is applied in order to identify the
Hidden surface(visible surface).


Figure 3 : Example 2 Scan Line Algorithm
1. First, examine the scanline(S 1), whose, Active edge table (Aet)
contains [AD,BC,RS,PQ], and the flag is set to “on” for surface(S 1).
In addition it contains surface(S 2) and the flag set “off” for the region
between BX and RX as it’s an outer region of the polygon’s surface
and not to be projecte d at view -port(display devices). N ow drop the
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color -intensities of the corresponding surfaces into the frame
buffer(refresh buffer).
2. Then, process the scanline(S 2), whose, Active edge table (Aet)
contains [AD,BC,RS,PQ], and the flag is set to “on” for surface
(ABCD) and surface(PQRS). Both of the polygons surfaces are
overlapping each other . So for this overlapped region that which of
the surface intensity should be taken into account? calculate the depth
(Zmin) of both surface(S 1) and surface(S 2) of the overlapped portion, of
surface intensity . N ext,
If depth(S 1)>depth(S 2), then the Flag of surface S 1=” on” and intensity
of surface S 1 will be considered else S 2, now
Drop the color -intensities of the corresponding surfaces whose flag is
set to on into the frame buffer(refresh buffer).
3. As Scanline(S 3) is passing through the same portion from where
Scanline(S 2) is passing, S 3 also has the same Active edge table(A et)
components as S 2 has and no need to calculate the depth(S1) and
depth(S2) again so S 3 can take the advantage of the concept of
Coherence.

10.2.3 Z-Buffer Algorithm :
1. One of the simplest and commonly use image space approach to
eliminate hidden surface is the Z-buffer or depth buffer algorithm. It
is developed by Catmull. This algorithm compares surface depths at
each pixel position on the projection plane.
2. The surface depth is measured from the view plane along the z axis of
a viewing system. When object desc ription is converted to projection
coordinates (x, y, z), each pixel position on the view plane is specified
by x and y coordinate, and z values gives the depth information. Thus,
object depths can be compared by comparing the z - values.
3. The Z -buffer algor ithm is usually implemented in the normalized
coordinates, so that z values range from 0 at the back -clipping plane
to 1 at the front clipping plane. The implementation requires another
buffer called Z -buffer along with the frame buffer memory required
for raster display devices. A Z -buffer is used to store depth values for
each (x, y) position as surfaces are processed, and the frame buffer
stores the intensity values for each position.
4. At the beginning the z -value at the back -clipping plane, and the frame
buffer is initialized to the background colour. Each surface listed in
the display file is then processed, one scan line at a time calculating
the depth (z -value) at each (x, y) pixel position.
5. The calculated depth value is compared to the value previousl y stored
in the Z -buffer at that position. If the calculated depth values is greater
that the value stored in the Z -buffer, the new depth value is stored,
and the surface intensity at that position is determined and placed in
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6. For example, among three surfaces, surface S 1 has the smallest depth
at view position (x, y) and the highest z value. So, it is visible at that
position.


Figure 4 : Example 1 Scan Line Algorithm
Algorithm :
1. Initialize the Z -buffer and frame buffer so that for all buffer positions
z-buffer(x,y)=0 and Frame -buffer(x,y)=Ibackground
2. During scan conversion process, for each position on each polygon
surface, compare depth values to previously stored values in the depth
buffer to determine visibility.
3. Calculate z -value for each (x,y) position on the polygon
If z>z -buffer(x,y) then
z-buffer(x,y)=z,
Frame -buffer(x,y)= Isurface(x,y).
4. Stop.
(After processing of all surfaces, the z -buffer contains depth values
for the
visible surfaces and the frame buffer contain the corresponding
intensity
values for those surfaces.)

Advantages :
1. It is easy to implement
2. It can be implemented in hardware to overcome the speed problem.
3. Since the algorithm processes objects one at a time, the total number
of polygons in a picture can be arbit rarily large.

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Disadvantages :
1. It requires an additional buffer and hence the large memory.
2. It is time consuming process as it requires comparison for each pixel
instead of for the entire polygon.

10.2.4 Warnock’s Algorithm (Area Subdivision Algorithm) :
An interesting approach to the hidden -surface problem was developed by
Warnock. He Developed area subdivision algorithm which subdivides
each area into four equal squares.

At each stage in the recursive -subdivision process, the relationship
between projection of each polygon and the area of interest is checked for
four possible relationships:
1. Surrounding polygon – One that completely encloses the (shaded)
area of interest.
2. Overlapping or Intersecting Polygon –One that is partly inside and
partly outside the area.
3. Inside or Contained Polygon – One that is completely inside the area.
4. Outside or Disjoint Polygon - One that is c ompletely outside the area.

Figure 5 : Area Subdivision Algorithm

After checking four relationship we can handle each relationship as
follows:
1. If all the polygons are disjoint from the area, then the background
colour is displayed in the area.
2. If there is only one intersecting or only one contained polygon, then
the area is first filled with background colour, and then the part of the
polygon contained in the area is filled with colour of polygon.
3. If there is a single surrounding polygon, but no intersecting or
contained polygons, then the area is filled with the colour of the
surrounding polygon.
4. If there are more than one polygon intersecting, contained in, or
surrounding the area then we must do some more processing.
Fig a) The four intersections of surrounding polygon are all closer to the
viewpoint than any of the other intersections. Therefore, the entire area is
filled with the colour of the surrounding polygon.
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b) That surrounding polygon is not completely in front of the i ntersecting
polygon.
The Warnock’s algorithm stops subdivision of area only when the problem
is simplified or even area is only a single pixel.

Algorithm:
1. Initialize the area to be the whole screen.
2. Create the list of polygons by sorting them with their z -values of
vertices. Don’t include disjoint polygons in the list because they are
not visible.
3. Find the relationship of each polygon.
4. Perform the visibility decision test
a. If all the polygons are disjoint from the area, then fill area with
background colour.
b. If there is only one intersecting or only one contained polygon
then first fill entire area with background colour and then fill the
part of the polygon contained in the area with the colour of
polygon.
c. If there is a single surrounding polygon, but no intersecting or
contained polygons, then fill the area with the colour of the
surrounding polygon.
d. If the surrounding polygon is closer to the viewpoint than all
other polygons, so that all other polygons are hidden by it, fill the
area with the colour of the surrounding polygon .
e. If the area is the pixel (x, y) and neither a, b, c, nor d apply,
compute the z coordinate at pixel (x, y) of all polygons in the list.
The pixel is then set to colour of the polygon which is closer to
the viewpoint.
5. If none of the above tests are true, then subdivide the area and go to
step 2.

Advantages :
1. It follows the divide -and-conquer strategy; therefore , parallel
computers can be used to speed up the process.
2. Extra memory buffer is not required.

10.2.5 Binary Space Partition (BSP) Trees :
Binary space partitioning is used to calculate visibility. To build
the BSP trees, one should start with polygons and label all the edges.
Dealing with only one edge at a time, extend each edge so that it splits the
plane into two. Place the f irst edge in the tree as root. Add subsequent
edges based on whether they are inside or outside. Edges that span the
extension of an edge that is already in the tree are split into two and both
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Figure 6: BSP trees
 From the above figure, first take A as a root.
 Make a list of all nodes in figure (a).
 Put all the nodes that are in front of root A to the left side of
node Aand put all those nodes that are behind the root A to the right
side as shown in figure (b).
 Process all the fro nt nodes first and then the nodes at the back.
 As shown in figure (c), we will first process the node B. As there is
nothing in front of the node B, we have put NIL. However, we have
node C at back of node B, so node C will go to the right side of
node B.
 Repeat the same process for the node D.

10.2.6 Painter’s Algorithm ( Depth Sorting Method ):
 Depth sorting method uses both image space and object -space
operations. The depth -sorting method performs two basic functions −
 First, the surfaces are sorted in order of decreasing depth.
 Second, the surfaces are scan -converted in order, starting with the
surface of greatest depth.
 The scan conversion of the polygon surfaces is performed in image
space. This method for solving the hidden -surface problem is oft en
referred to as the painter's algorithm. The following figure shows the
effect of depth sorting −

Figure 7: Depth Sorting Method
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 The algorithm begins by sorting by depth. For example, the initial
“depth” estimate of a polygon may be taken to be the closest z value
of any vertex of the polygon.
 Let us take the polygon P at the end of the list. Consider all polygons
Q whose z -extents overlap P’s. Before drawing P, we make the
following tests. If any of the following tests is positive, then w e can
assume P can be drawn before Q.
 Do the x -extents not overlap?
 Do the y -extents not overlap?
 Is P entirely on the opposite side of Q’s plane from the viewpoint?
 Is Q entirely on the same side of P’s plane as the viewpoint?
 Do the projections of the po lygons not overlap?
 If all the tests fail, then we split either P or Q using the plane of the
other. The new cut polygons are inserting into the depth order and the
process continues. Theoretically, this partitioning could generate
O(n2) individual polygon s, but in practice, the number of polygons is
much smaller.

10.2.7 Visible -Surface Ray Tracing :
 Ray tracing is an image -based algorithm. For every pixel in the image,
a ray is cast from the center of projection through that pixel and into
the scene. The colour of the pixel is set to the colour of the object that
is first encountered.

Figure 8: Visible -Surface Ray Tracing

10.3 COMPARISON OF THE METHODS
 Hardware available? Use depth -buffer, possibly in combination with
back -face elimination or depth -sort for part of scene.
 If not, choose dependent on complexity scene and type of objects:
 Simple scene, few objects: depth -sort
 Quadratic surfaces: ray -casting
 Otherwise: depth -buffer
 Many additional methods to boost performance (kD -trees, scene
decomposition, etc.)

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10.4 SUMMARY
When we view a picture containing non -transparent objects and surfaces,
then we cannot see those objects from view which are behind from objects
closer to eye. We must remove these hidden surfaces to get a realistic
screen image. The identification and removal of these surfaces is
called Hidden -surface problem. We learn many algo rithms to remove
hidden surface.

10.5 QUESTIONS
Q1. Explain Scan -Line Method.
Q2. Write note on Area -Subdivision Method.
Q3. Explain Binary Space Partition (BSP) Trees with example
Q4. What is the concept of Back Face Culling? Explain.
Q7. Explain Painters algorithm.
Q8. Differentiate between Ray Casting and R ay Tracing.
Q9. Describe the methods used for determination of visible surfaces in
rendering.
Q10. Explain techniques for efficient Visible -Surface Algorithms.
Q11. Explain Image Space Method.
Q12. Write a Short Note on Object Space Method.

10.6 REFERENCES
o Computer Graphics Hearn, Baker Pearson Education Second
o Steve Marschner, Peter Shirley Fundamentals of Computer Graphics
CRC press Fourth Edition 2016.
o A P Godse, Computer Graphics.




*****
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11

PLANE CURVES AND SURFACES

Unit Structure
11.0 Objectives
11.1 Curve Representation
11.1.1 Nonparametric Curves, Parametric Curves
11.2 Parametric Representation of a Circle
11.3 Parametric Representation of an Ellipse
11.4 Parametric Representation of a Parabola
11.5 Parametric Representation of a Hyperbola
11.6 Representation of S pace Curves
11.6.1 Bezier Curves
11.6.2 B-spline Curves
11.6.3 B-spline Curve Subdivision
11.6.4 Parametric Cubic Curves
11.6.5 Quadric Surfaces
11.6.6 Bezier Surfaces.
11.7 Summary
11.8 Questions
11.9 References
11.10 Multiple Choice Questions

11.0 OBJECTIVES
1. To learn curves and surfaces and types of curves.
2. To understand the parametric representation of Curves.
3. To understand Quadratic Surfaces and curves .
4. To understand Bezier curve/surface and their properties.

11.1 INTRODUCTION TO CURVE
In computer graphics, we often need to draw different types of objects
onto the screen. Objects are not flat all the time and we need to draw
curves many times to draw an object.

11.1.1 Types of Curves :
A curve is an infinitely large set of points. Each point has two neighbours
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Implicit Curves :
Implicit curve representations define the set of points on a curve by
employing a procedure that can test to see if a po int in on the curve.
Usually, an implicit curve is defined by an implicit function of the form –
f(x, y) = 0
It can represent multivalued curves (multiple y values for an x value). A
common example is the circle, whose implicit representation is
x2 + y2 - R2 = 0

Explicit Curves :
A mathematical function y = f(x) can be plotted as a curve. Such a
function is the explicit representation of the curve. The explicit
representation is not general, since it cannot represent vertical lines and is
also single -valued. For each value of x, only a single value of y is
normally computed by the function.
Parametric Curves :
Curves having parametric form are called parametric curves. The explicit
and implicit curve representations can be used only when the fun ction is
known. In practice the parametric curves are used. A two -dimensional
parametric curve has the following form −
P(t) = f(t), g(t) or P(t) = x(t), y(t)
The functions f and g become the (x, y) coordinates of any point on the
curve, and the points are obtained when the parameter t is varied over a
certain interval [a, b], normally [0, 1].

11.2 PARAMETRIC REPRESENTATION OF CURVE CIRCLE
The most common way to write a circle is x2+y2=x2
Parametrically, this can be written as
y(t) =r. sin(2. Pi. t)
x(t) =r. cos(2. Pi. t)
or if we state the angle in degrees:
y(t) =r. sin(360. t)
x(t) =r. cos(360. t)
when t runs from 0 to 1, the angle argument runs a full circle, and the
point (x(t),y(t)) describes a circle.

11.3 PARAMETRIC REPRESENTATION OF ELLIPSE
An elli pse can be defined as the locus of all the points that satisfy the
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Where x, y are the coordinates of any point on the ellipse, a, b are the
radius on the x and y axes respectively, t is the parameter which ranges
from 0 to 2𝜋 radians.
describe the ellipse as (x/rx)2 +(y/ry)2 =1
Parametrically we describe an ellipse like this:
x(t’) = r x . cos(t’)
y(t’) = r y . sin(t’)
0 ≤ t ≤ 2π

The parametric equations of an ellipse centered at the origin

Recall the construction of a point of an ellipse using two concentric circles
of radil equal to lengths of the semi – axes a and b, with the center at the
origin as shows the figure,then
𝑥=𝑎cos𝑡
𝑦=𝑏sin𝑡

Where, 0≤𝑡 <2𝑥

To convert the above parametric equations into Cartesian co -ordinates,
divide the first equation by a and the second by b then square and add
them.
𝑥ଶ
𝑎ଶ+𝑦ଶ
𝑏ଶ=1
Thus, obtained is the standard equation of the ellipse.

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The parametric equations of a tra nslated ellipse with center at (𝑥0𝑦0)
The parametric equations of a translated ellipse with center (𝑥0𝑦0) and
semi -axes a and b,
𝑥=𝑥0+𝑎cos𝑡
𝑦=𝑦0+𝑏sin 𝑡
Where, 0≤𝑡<2𝜋
To convert the above param etric equations into Cartesian co -ordinates, we
write them as
𝑥−𝑥0=𝑎 𝑐𝑜𝑠 𝑡
𝑦−𝑦0=𝑏sin 𝑡

And divide the first equation by a and second by b, then square and add
both equations, so we get
(௫ି௫బ)ଶ
௔మ+(௬ି௬బ)ଶ
௕మ=1


11.4 PARAMETRIC REPRESENTATION OF PARABOLA
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Since every parabola is congruent to x2 = 4ay, for some choice of a> o, we
can study the geometry of every parabola by confining ourselves to this
one. This shows the power of both coordinate geometry and
transformations. Imagine a particle moving in the plane along the curve C
as shown.
The x - and y - coordinate of the particle can be thought of as functions of a
new variable t, and so we can write x=f(t) , y= g(t), where f, g are functions
of t. t is time.
The equations x= f(t), y = g(t) are called the parametric equations of the
point P(x,y) and the variable t is called a parameter.
It is often very useful to take a Cartesian equation y= F(x) and introduce a
parameter t so that the x and y – coordinates of any point on the curve can
be expressed In terms of this parameter.

11.5 PARAMETRIC REPRESENTATION OF HYPERBOLA
A hyperbola centred at the origin with a left -right opening has the
equation:
(x2/a2) - (y2/b2) =1
In this case, we can use the equations x= a sec t, y= b tan t. This is because
if you plug these expressions in to the above equation, then the equation
reduces to the trigonometric identity sec2 t – tan2 t =1
Hyperbola centred at the origin with an up -down opening has the equation:
(y2/a2) - (x2/b2) =1
In this case, switch the expressions for x and y, using the equations
X= b tan t, y= a sec t.
If the center of the hyperbola is at (h, k), then the equation is:
((x-h)2/a2) - ((y-k)2/b2) =1 or ((y -k)2/a2) - ((x-h)2/b2) =1
In the first case, the parametric equation x - h _a sec t, y= K + b tan t will
work, and in the second case it will be x -h + b tan t, y=k +a sec t.

11.6 REPRESENTATION OF SPACE CURVES
11.6.1 Bezier curve :
Bezier curve is discovered by the French engineer Pierre Bézier . These
curves can be generated under the control of other points. Approximate
tangents by using control points are used to generate curve. The Bezier
curve can be represented mathematically as −
n∑k=0 PiB in(t)
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Bin(t)= (n i) (1−t)n−tti
Where n is the polynomial degree, i is the index, and t is the variable.
The simplest Bezier curve is the straight line from the point P0 to P1. A
quadratic Bezier curve is determined by three control points. A cubic
Bezier curve is determined by four control points.

Figure 1: Bezier Curves

11.6.2. Properties of Bezier Curves :
Bezier curves have the following properties :
 They generally follow the shape of the control polygon, which
consists of the segments joining the control points.
 They always pass through the first and last control points.
 They are contained in the convex hull of their def ining control points.
 The degree of the polynomial defining the curve segment is one less
that the number of defining polygon point. Therefore, for 4 control
points, the degree of the polynomial is 3, i.e. cubic polynomial.
 A Bezier curve generally follows the shape of the defining polygon.
 The direction of the tangent vector at the end points is same as that of
the vector determined by first and last segments.
 The convex hull property for a Bezier curve ensures that the
polynomial smoothly follows the cont rol points.
 No straight line intersects a Bezier curve more times than it intersects
its control polygon.
 They are invariant under an affine transformation.
 Bezier curves exhibit global control means moving a control point
alters the shape of the whole curve.
 A given Bezier curve can be subdivided at a point t=t0 into two Bezier
segments which join at the point corresponding to the parameter value
t=t0.


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11.6.3. Applications of Bezier Curves :
Bezier curves have their applications in the following fields -
1. Computer Graphics :
 Bezier curves are widely used in computer graphics to model smooth
curves.
 The curve is completely contained in the convex hull of its control
points.
 So, the points can be graphically displayed & used to ma nipulate the
curve intuitively.
2. Animation :
 Bezier curves are used to outline movement in animation applications
such as Adobe Flash and synfig.
 Users outline the wanted path in bezier curves.
 The application creates the needed frames for the object to move
along the path.
 For 3D animation, bezier curves are often used to define 3D paths as
well as 2D curves.
3. Fonts :
True type fonts use composite bezier curves composed of quadratic bezier
curves.
 Modern imaging systems like postscript, asymptote etc use composite
bezier curves composed of cubic bezier curves for drawing curved
shapes.

11.6.4. Given a bezier curve with 4 control points - B0[1 0] , B1[3 3] ,
B2[6 3] , B3[8 1] Determine any 5 points lying on the curve.
Solution: We have -
 The given curve is defined by 4 control points.
 So, the given curve is a cubic bezier curve.
The parametric equation for a cubic bezier curve is -
P(t) = B0(1 -t)3 + B13t(1 -t)2 + B23t2(1 -t) + B3t3
Substituting the control points B0, B1, B2 and B3, we get -
P(t) = [1 0](1 -t)3 + [3 3]3t(1 -t)2 + [6 3]3t2(1 -t) + [8 1]t3 ……..(1)
Now,
To get 5 points lying on the curve, assume any 5 values of t lying in the
range 0 <= t <= 1.
Let 5 values of t are 0, 0.2, 0.5, 0.7, 1
For t = 0:
Substituting t=0 in (1), we get -
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P(0) = [1 0] + 0 + 0 + 0
P(0) = [1 0]
For t = 0.2:

Substituting t=0.2 in (1), we get -
P(0.2) = [1 0](1 -0.2)3 + [3 3]3(0.2)(1 -0.2)2 + [6 3]3(0.2)2(1 -0.2) + [8
1](0.2)3
P(0.2) = [1 0](0.8)3 + [3 3]3(0.2)(0.8)2 + [6 3]3(0.2)2(0.8) + [8 1](0.2)3
P(0.2) = [1 0] x 0.512 + [3 3] x 3 x 0.2 x 0.64 + [6 3] x 3 x 0.04 x 0.8 + [8
1] x 0.008
P(0.2) = [1 0] x 0.512 + [3 3] x 0.384 + [6 3] x 0.096 + [8 1] x 0.008
P(0.2) = [0.512 0] + [1.152 1.152] + [0.5 76 0.288] + [0.064 0.008]
P(0.2) = [2.304 1.448]
For t = 0.5:

Substituting t=0.5 in (1), we get -
P(0.5) = [1 0](1 -0.5)3 + [3 3]3(0.5)(1 -0.5)2 + [6 3]3(0.5)2(1 -0.5) + [8
1](0.5)3
P(0.5) = [1 0](0.5)3 + [3 3]3(0.5)(0.5)2 + [6 3]3(0.5)2(0.5) + [8 1](0.5)3
P(0.5) = [1 0] x 0.125 + [3 3] x 3 x 0.5 x 0.25 + [6 3] x 3 x 0.25 x 0.5 + [8
1] x 0.125
P(0.5) = [1 0] x 0.125 + [3 3] x 0.375 + [6 3] x 0.375 + [8 1] x 0.125
P(0.5) = [0.125 0] + [1.125 1.125] + [2.25 1.125] + [1 0.125]
P(0.5) = [4.5 2.375]
For t = 0.7:

Substituting t=0.7 in (1), we get -
P(t) = [1 0](1 -t)3 + [3 3]3t(1 -t)2 + [6 3]3t2(1 -t) + [8 1]t3
P(0.7) = [1 0](1 -0.7)3 + [3 3]3(0.7)(1 -0.7)2 + [6 3]3(0.7)2(1 -0.7) + [8
1](0.7)3
P(0.7) = [1 0](0.3)3 + [3 3]3(0.7)(0.3)2 + [6 3]3(0.7)2(0.3) + [8 1](0.7)3
P(0.7 ) = [1 0] x 0.027 + [3 3] x 3 x 0.7 x 0.09 + [6 3] x 3 x 0.49 x 0.3 + [8
1] x 0.343
P(0.7) = [1 0] x 0.027 + [3 3] x 0.189 + [6 3] x 0.441 + [8 1] x 0.343
P(0.7) = [0.027 0] + [0.567 0.567] + [2.646 1.323] + [2.744 0.343]
P(0.7) = [5.984 2.233]
For t = 1:

Substituting t=1 in (1), we get -
P(1) = [1 0](1 -1)3 + [3 3]3(1)(1 -1)2 + [6 3]3(1)2(1 -1) + [8 1](1)3
P(1) = [1 0] x 0 + [3 3] x 3 x 1 x 0 + [6 3] x 3 x 1 x 0 + [8 1] x 1
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P(1) = [8 1]


Figure 2: Example of Bezier Curve

11.6.2 B-Spline Curves :
The Bezier -curve produced by the Bernstein basis function has limited
flexibility.
 First, the number of specified polygon vertices fixes the order of the
resulting polynomial which defines the curve.
 The second limiting characteristic is that the v alue of the blending
function is nonzero for all parameter values over the entire curve.
The B -spline basis contains the Bernstein basis as the special case. The B -
spline basis is non -global.
A B-spline curve is defined as a linear combination of control p oints Pi
and B -spline basis function Ni, k (t) given by Ni, k
C(t)=∑n
i=0P, Ni,k(t), n≥k−1, tϵ[tk−1,tn+1]
Where,
 {pi: i=0, 1, 2….n} are the control points
 k is the order of the polynomial segments of the B -spline curve. Order
k means that the curve is made up of piecewise polynomial segments
of degree k - 1,
 the Ni,k(t) are the “normalized B -spline blending functions”. They are
described by the order k and by a non -decreasing sequence of real
numbers normally called the “knot sequence”.
ti:i=0,... n+K
The N i, k functions are described as follows −
Ni,1(t)={1, ifuϵ[ti,ti+1)
0,Otherwise
and if k > 1,
Ni,k(t)=(t−ti/ti+k−1) Ni,k−1(t)+ (ti+k−t/ti+k−ti+1) Ni+1,k−1(t)
and
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tϵ[tk−1,tn+1)

Properties of B -spline Curve :
B-spline curves have the following properties:
 The sum of the B -spline basis functions for any parameter value is 1.
 Each basis function is positive or zero for all parameter values.
 Each basis function has precisely one maximum value, except for
k=1.
 The maximum order of the curve is equal to the number of vertices of
defining polygon.
 The degree of B -spline polynomial is independent on the number of
vertices of defining polygon.
 B-spline allows the local co ntrol over the curve surface because each
vertex affects the shape of a curve only over a range of parameter
values where its associated basis function is nonzero.
 The curve exhibits the variation diminishing property.
 The curve generally follows the shape of defining polygon.
 Any affine transformation can be applied to the curve by applying it
to the vertices of defining polygon.
 The curve line within the convex hull of its defining polygon

11.6.3 B-Spline Curve Subdivision :
Subdivision “Subdivision defines a smooth curve or surface as the limit of
a sequence of successive refinements ”
The sub division of B - Spline curve used De Boor’s Algorithm. It provides
a fast and numerically stable way for finding a point on a B -spline c urve
given unit domain. It is the algorithm for evaluating spline curves in B -
spline forms.

Selecting Control Points :
Suppose we want to subdivide a B -spline curve at u into two B -spline
curves, one on [0,u] and the other on [u,1]. The first thing to do is to apply
de Boor's algorithm at u. Note that if u is in [uk, uk+1), then at most p+1
control points will be used in the computation, namely: Pk, Pk-1, ..., Pk-p.
To find the control points of the B -spline curve on [0,u], we start
with P0 and follow the control polyline. When we encounter a control
point or a point computed in the process of de Boor's algorithm, we select
that point and make a turn. This process continues until we reach C(u)
which is also selected as a control point. These points, in the visiting
order, form the control points that define the B -spline curve on [0,u]. For
the B -spline curve defined on [u,1], we start with the point C(u) and apply
the same process. We select each encountered point and make a turn
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Figu re 3: Plotting of Control Points

11.6.4 Parametric Cubic curves :
Cubic curves are commonly used in graphics because curves of lower
order commonly have too little flexibility, while curves of higher order are
usually considered unnecessarily complex and make it easy to introduce
undesired wiggles. A parametric cubic curve in 3D is defined by: usuall y ,
we consider t = [0…1].

Figure 4: Cubic Curves
11.6.5 Quadratic Surfaces :
A second order algebraic surface given by the general equation:
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Examples of quadratic surfaces include the cone, cylinder, ellipsoid,
elliptic cone, elliptic cylinder, hyperbolic cylinder, hyperbolic parabo loid,
paraboloid, sphere and spheroid.

Figure 5: Quadratic Surfaces

11.6.6 Bezier surface :
Bezier surfaces are a species of mathematical spline used in computer
graphics , computer -aided design, and finite element modeling.
 Bezier surface are an extension of the idea of Bezier curves and share
many of their properties.
 Bezier surface is defined by a set of control points.
 Properties are -
o the surface does not in general pass through the control points (except
for the corner points)
o the surface is contained within the convex hull of the control points.
A two -dimensional Bézier surface can be defi ned as a parametric
surface where the position of a point p as a function of the parametric
coordinates u, v is given by :

evaluated over the unit square , where

is a Bernstein polynomial , and
is the binomial
coefficient .
Some properties of Bézier surfaces:
 A Bézier surface will transform in the same way as its control points
under all linear transformations and translations .
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 All u = constant and v = constant lines in the ( u, v) space, and all four
edges of the deformed ( u, v) unit square are Bézier curves.
 A Bézier surface will lie completely within the convex hull of its
control points, and therefore also completely within the bounding
box of its control points in any given Cartesian coordinat e system .
 The points in the patch corresponding to the corners of the deformed
unit square coincide with four of the control points.
 However, a Bézier surface does not generally pass through its other
control points.

Figure 6: Bézier surface
Sample Bézier surface; red - control points, blue - control grid, black -
surface approximation

11.7 SUMMARY
When we view a picture containing non -transparent objects and surfaces,
then we cannot see those objects from view which are behind from objects
closer to eye. We must remove these hidden surfaces to get a realistic
screen image. The identification and removal of these surfaces is
called Hidden -surface problem. We learn many algorithms to remove
hidden surface and along wit h that different types of curves and surface
with its types.

11.8 QUESTIONS
Q1. Explain Parametric Representation of a Circle .
Q2. Explain Parametric Representation of an Ellipse .
Q3. Explain Parametric Representation of a Parabola and Hyperbola
Q4. Explain Bezier Curves and surfaces.
Q5. Define Curve. What are its types? Explain in brief.
Q6. What are the properties of Bezier Curves? Explain.
Q7. Explain Bezier Curves properties .
Q9. Write a short note on Quadratic Surfaces.
Q10. Explain Bezier Su rfaces with example.


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11.9 REFERENCES 1. Computer Graphics Hearn, Baker Pearson Education Second
2. Steve Marschner, Peter Shirley Fundamentals of Computer Graphics
CRC press Fourth Edition 2016.
3. A P Godse, Computer Graphics.

11.10 MULTIPLE CHOICE QUESTIONS
1. In __________ method, visibility is decided point by point at each
pixel position on the projection plane.
a. Image -Space
b. Object -Space
c. Polygon Clipping
d. Coherence
2. Z-buffer algorithm is used for ___________.
a. Frame buffer removal
b. Hidden line removal
c. Rendering
d. Animation
3. ______________ is fast and effective as an initial screening to
eliminate many polygons from further visibility tests.
a. Back -face detection
b. Painter’s algorithm
c. Z-buffer algorithm
d. Area Sub -division algorithm
4. The _________ is particularly useful when the view reference point
changes, but the objects in a scene are at fixed positions.
a. BSP Tree
b. Back -face
c. Painter’s algorithm
d. Z-buffer algorithm
5. ______ method successively divides the total viewing area into
smalle r and smaller rectangles until each small area is the projection
of part of a single visible surface or no surface at all.
a. Area -subdivision method
b. Back -face
c. Painter’s algorithm
d. Z-buffer algorithm
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c. Scan line algorithm
d. Ray Tracing
7. Most visible -surface detection algorithms use sorting and __________
methods to improve performance.
a. Coherence
b. Object -Space
c. Polygon Clipping
d. Dept information
8. Visible Surface Detecti on Method is also called as _____________.
a. Hidden -Surface elimination method
b. Hidden - Surface Identification Method
c. Hidden -Surface Detection Method
d. VSD
9. Z-buffer algorithm uses 2 buffer, depth buffer for storing depth values
and _________ buffer stores in tensity values at each pixel position.
a. Refresh
b. Intensity
c. Color
d. Depth
10. ________ curves must precisely have n control points.
a. Bezier
b. B-Spline
c. Curved
d. Parametric

*****
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UNIT V

12
COMPUTER ANIMATION


Unit Structure
12.0 Objectives
12.1 Introduction
12.2 Definition
12.3 Principles of Animation
12.3.1 Squash and Stretch
12.3.2 Anticipation
12.3.3 Staging
12.3.4 Straight Ahead Action and Pose to Pose
12.3.5 Follow Through and Overlapping Action
12.3.6 Ease In and Ease Out
12.3.7 Arc
12.3.8 Secondary Action
12.3.9 Timing
12.3.10 Exaggeration
12.3.11 Solid drawing
12.3.12 Appeal
12.4 Key Frame Animati on
12.5 Deformation
12.6 Character Animation
12.7 Physics Based Animation
12.8 Procedural Techniques
12.9 Groups Of objects
12.10 Summary
12.11 References
12.12 Bibliography
12.13 Exercise

12.0 OBJECTIVES
After this chapter, student will be able to:
 Understand the computer animation and examples of animation .
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 Study the types of animation like key frame animation, character
animation, Physics based animation.
 Understand the process of deformation and its requirement.
 Understand the concept behind object groups.

12.1 INTRODUCTION
 Computer animation is growing industry in the area of Web, TV,
games , multimedia , films, digital art, architectural visualization etc.
 The goal of computer animation is to process the desired motion effect
by mixing perception, natural inform ation and imagination.
 It helps us to understand physical law by adding motion control to data.
 Many people can use computer animation in profession like pilots,
architect, surgeons, automobile company, gaming industry etc.

12.2 DEFINITION
 Computer animation may be defined as a technique in which the
illusion of a movement is created by displaying on a screen or
recording on device individual states of a dynamic scene.
 Any computer animation sequence may be defined as a set of objects.
 It is characterized by state variables evolving over time or it is a time
sequence of changes in a scene.
 For example, a human character is normally characterized using its
joint angles as state variables.
 Animation is production of consecutive images whi ch when displayed
together convey a feeling of motion.
 In addition to changing object position with translations or rotations, a
computer generated animation could display time variations in object
size, color, texture etc.
 Computer animations can also b e generated by changing camera
parameters such as position, orientation and focal length.



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12.3 PRINCIPLES OF ANIMATION
Principles of animation can be categorized in following types:

12.3.1 SQUASH AND STRETCH:

 The most important pr inciple is "squash and stretch".
 The purpose of this is to give a sense of weight and flexibility to drawn
objects.
 It ca n be applied to simple objects, example like a bouncing ball,
construction like the musculature of a human face etc.
 A figure stretched or squashed to an exaggerated degree can have a
comical effect.
 In realistic animation, the most important aspect of this principle is the
fact that an object's volume does not change when squashed or
stretched.
 If the length of a ball is stretched vertically, its width (in three
dimensions and also its depth) needs to contract correspondingly
horizontally.



12.3.2 ANTICIPATION :
 Anticipation is used to prep are the audience for an action.
 It make s the action appear more realistic.
 A dancer jumping off the flo or has to bend the knees first, a golfer
making a swing has to swing the club back first are example of it .





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12.3.3 STAGING :
 Its purpose is to direct the audience's attention.
 This makes it clear that what is of greatest importanc e in a scene.
 Johnston and Thomas defined it as "the presentation of any idea so that
it is completely and unmistakably clear", whether that idea is an action ,
a personality, an expression or a mood.
 This can be done by various means such as the placement of a character
in the fram e, the use of light and shadow or the angle and position of
the camera.
 The essence of this principle is ke eping focus on what is relevant .



12.3.4 STRAIGHT AHEAD ACTIO N AND POSE TO POSE :

These are two different approaches to the actual drawing process.
 "Straight ahead action" scenes are animated frame by frame from
beginning to end.
 Straight ahead action creates a more flui d, dynamic illusion of
movement.
 It is better for producing realistic action sequences.
 It is hard to maintain proportions and to create exact, convincing poses
along the way.
 "Pose to pose" involves startin g with drawing a few key frames and
then filling in the intervals later.
 "Pose to pose" works better for dramatic or emotional scenes.
 Here composition and relation to the surroundings are of greater
importance.
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12.3.5 FOLLOW THROUGH AND O VERLAPPING ACTION :

Follow through and overlapping action is a general heading for two
closely related techniques.
 It helps to render movement more realistically.
 "Follow through" means that loosely tied parts of a body should
continue moving aft er the character has stopped.
 Also the parts should keep moving beyond the point where the
character stopped only to be subsequently "pulled back towards
the center of mass or exhibiting various degree s of oscillation damping.
 "Overlapping action" is the tendency for parts of the b ody to move at
different rates example an arm will move on differe nt timing of the
head and so on .
 A third related technique is "drag" where a character starts to move and
parts of him take a few frames to catch up.
 These parts can be inanimate objects like cl othing or the antenna on a
car or parts of the body, such as arms or hair.



12.3.6 EASE IN AND EASE OUT :

 The movement of the human body and most other objects needs time to
accelerate and slow down.
 For this reason, animation looks more realistic .
 It has more drawings near the beginning and end of an action,
emphasizing the extreme poses and fewer in the middle.
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 This principle goes for characters mo ving between two extreme poses
for example sitting down and standing up but also for inanimate
moving objects like the bouncing ball.


12.3.7 ARC:

 Most natural action tends to follow an arched trajectory and animation
should adhere to this principle by following implied "arcs" for greater
realism.
 This technique can be applied to a m oving limb by rotating a joint or a
thrown object moving along a parabolic trajector y.
 As an object's speed or momentum increases, arcs tend to flatten out in
moving ahead an d broaden in turns.
 An object in motion that moves out of its natural arc for no apparent
reason will appear erratic rather than fluid.
 For example, when animating a pointing finger, the animator should be
certain that in all drawings i n between the two e xtreme poses the
fingertip follows a logical arc from one extreme to the next.



12.3.8 SECONDARY ACTION:

 Adding secondary actions to the main action gives a scene more li fe.
 It can help to support the main action.
 A person walking can simultaneously swing their arms or keep them in
their pockets, speak or whistle or express emotions through facial
expressions.
 The important thing about secondary actions is that they emphasize
rather than take attention away from the main action.
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 For example during a dramatic movement, facial expressions will often
go unnoticed.
 In these cases it is better to include them at the beginni ng and the end
of the movement rather than during.



12.3.9 TIMING :

 Timing refers to the number of drawing s or frames for a given action.
 It translates to the speed of the action on film.
 Correct timing makes objects appe ar to obey the laws of physics.
 Example an object's weight determin es how it reacts to an impetus like
a push.
 Timing i s critical for establishi ng a character's mood, emotion and
reaction.
 It can also be a device to communicate aspects of a character's
personality .



12.3.10 EXAGGERATION :

 Exaggeration is an effect e specially useful for animation.
 It enhances animated motions that strive for a perfect imitation of
reality can look static and dull.
 The level of exaggeration depends on whether one seeks realism or a
particular style like a caricature or the style of a specific artist.
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 The class ical definition of exaggeration emp loyed by Disney was to
remain true to reality just presenting it in a wilder and in more extreme
form.
 If a scene contains several elements, there should be a balance in how
those elements are exagger ated in relation to each other.
 This is to avoid confusi ng or overawing the viewer.



12.3.11 SOLID DRAWING :

 The principle of solid drawing means taking into account fo rms in
three -dimensional space or giving them volume and weight.
 The animator n eeds to be a skilled artist.
 He has to understand the basics of three -dimensional shapes, anatomy,
weight, balance, light and shadow etc.


12.3.12 APPEAL :

 Appeal in a cartoon character corresponds to what would be
called charisma in an actor.
 A character that is appealing i s not necessarily sympathetic,
 Villains or monsters can also be appealing.
 The important thing is that the viewer feels the character is real and
interesting.
 There is way for making a character connect better with the audience .
 Example a complicated or hard to read face w ill lack appeal.
 It may more accurately be described as 'captivation' in the composition
of the pose or the character design.

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12.4 KEY-FRAME ANIMATION
 A key frame animation consists of an automatic generation of the
intermediate fram es based on a set of key frames that is supplied by an
animator.
 A key frame in animation and filmmaking is a drawing that defines the
starting and ending points of any smooth transition.
 In order to produce a key -frame animation the animator creates the
behavior of a model manually by using an intuitive the “put that there”
methodology.
 The animator has full and direct control over the positions, shapes and
motions of models during the animation.
 There are two fundamental approaches to a key -frame:
1. In-betweens are obtained by shape interpolation.
This method makes it possible to transform a geometrical form into
another during an animation.
There is a serious probl em for an image based key frame as the
motion can be distorted due to the interpolati on.
The problem becomes complicated when the two drawings do not
have the same number of vertices.
In such case it is initially necessary to carry out a preprocessing step
to equalize the number of vertices of the two drawings.
2. A way of producing better images is to interpolate parameters of the
model instead of the object itself.
The parameters are normally spatial parameters, physical parameters
and visualization parameters that decide the model's behavior.

12.5 DEFORMATION
 It’s an important aspect of geometry processing used in CA D and
movie industry.
 In this shape manipulation is driven by a variation optimization.
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 It guarantees high quality deformations by minimizing physically
inspired energies subject to user controlled constraints.
 Physically -plausible deformations are achieved by a constrained
minimization of stretching (center) or bending (right) energies.
 In defo rmations on a very large model user deforms the shape using
reduced model.





12.6 CHARACTER ANIMATION:
 This type of animation defines specialized area of animation in which
animated character are brought to life.
 Character animators are often known as “actors with pencil”.
 Character animator captures the performance data with varies input
technology using camera, key board, microphone, mouse or touch
devices etc.
 They capture facial expression.
 This make animation look s real.
 Character animator tries to put life in their character work for creating
the illusion of emotion, personality and thoughts.
 Example of this are animated feature films, gaming industry,

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12.7 PHYSICS BASED ANIMATION
 Physics based animation is study of simulation of physically plausible
behaviors at collaborative rate.
 This animation is constraint by strict time as it is imposed by interactive
animation.
 It is inspired by need to include physically motivated behaviors in
complex movie, video games etc.
 In these animation an animator provides physical data and the motion is
obtained by solving the dynamics equations.
 It focuses on numerical stability and visual appeal over the physical
accuracy.
 Here m otion is globally controlled.
 We can distinguish methods based on parameter like adjustment and
constraint -based methods .
 Here the animator states are in terms of constraints and the properties
the model is supposed to have without needing to adjust parameters.

12.8 PROCEDURAL TECHNIQUES
 A procedural technique helps a type of computer animation which is
used to automatically generate animation in real time.
 It allow for a more diverse series of actions than could also be created
using predefined animations.
 The proced ural technique is us ed to simulate particle system like
smoke, fire, water, etc. , cloth and clothing, rigid -body dynamics, hair
and fur dynamics and character animation.
 In computer and video games it is often used for simple things like
turning a character's head when a player looks around.
 The procedural techniques correspond to the creat ion of a motion by a
procedure by describing the motion.
 Rule s are established for a system and an initial state is defined for
objects in the system.
 The procedural technique can be very useful for generating much life -
like motion from relatively little input.
 Such an animation should be used when the motion can be described by
an algorithm or a formula.
 The generation of a motion using a pr ocedure is not really a method but
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 There are so many possibilities of expressing a motion using a
procedure or a law.
 The number of all possible algorithms is unlimited.
 Procedural approaches can suffer from a lack of explicit control over
the look of individual frames.

12.9 GROUPS OF OBJECT
 We should know how to group a set of objects together to apply a
single animation.
 To animate multiple objects you have to group them first.
 First right click the selected object and then click on Group > Group.
 Next on animation tab, click on effect that you want to apply to group.
 Apply your animation effect only once.
 Next once finished, when you play back the animation the group will
behave like one object.
 If you want to ungroup the objects, click on ungroup option.
 There is also another way of grouping the objects called as drawing a
marquee.
 Example in PowerPoint you can animate the group of objects such as
clip arts, shapes and pictures.
 This helps to draw audience attention to specific content or to make it
easier to read.

12.10 SUMMARY
 Computer animation is art of creating moving images or objects with
the help of comput er.
 Computer animation is the field area of computer graphics.
 Animator is able to tell story layout and communicate emotions and
ideas by applying the principles of animation.
 The simplest form of animating the objects can be done by key frame
animation.
 Animation consist of many types such as character animation, key
frame animation, physics based animation and so on.
 Procedural techniques are used to generate procedural animation.
 Techniques like deformation and grouping of objects are used in
animation to make a study easy.
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12.11 EXERCISE
1. Write a short not on key frame animation.
2. Define computer animation. State and explain principles of animation.
3. Explain the concept of deformation.
4. State and explain procedural techniques in animation.
5. Write a short note on character animation and physics based
animation.
6. Explain the concept of how to group the objects.

12.12 REFERENCES
1. Computer Graphics , R. K. Maurya, John Wiley.

12.13 BIBLIOGRAPHY
1. D. Hearn and M. P. Baker, Computer Graphics , 2nd Editio n, 1997.
2. J. Hughes, A. Van Dam, M. MCGuire, D. Sklar, J. Foley, and K.
Akeley, Computer Graphics, Principles and Practice , 3rd Editio n,
Pearson, 2014.
3. P. Shirley and S. Marschner, “Fundamentals of Computer Graphics:
Ray Tracing,” CRC Press . p. 752, 2009.







*****












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UNIT V
13
IMAGE MANIPULATION A ND STORAGE

Unit Structure
13.0 Objectives
13.1 Introduction
13.2 What is an Image ?
13.2.1 Image Types
13.3 Digital Image File Formats
13.4 Image Compression
13.4.1 Image Compression Standard -JPEG
13.5 Image processing
13.5.1 Digital Image Enhancement
13.5.2 Contrast Stretching
13.5.3 Histogram
13.5.3.1 Histogram Equalization
13.5.4 Filtering
13.5.4.1 Smoothing
13.5.4.1.1 Low Pass Averaging Filter
13.5.4.1.2 Low Pass Median Filter
13.6 Summary
13.7 References
13.8 Bibliography
13.9 Exercise

13.0 OBJECTIVES After this chapter, student will be able to:
 Understand two-dimensional image and image processing system.
 Understand different image file format available.
 Study the image enhancement techniques to perform subjective
improvisation of images and filtering techniques in spatial domain.
 Understand image compression standard for optimization of storage.
 Understand histogram and how to find histogram of image by
equalization method.

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13.1 INTRODUCTION
 Human being depends on eye to capture the information that is around
them.
 Among the five senses human being have, we depend on sight the most.
 There is some example which shows that not every animal depend on
visual system as human being do.
 Example of such animals is bats which use high f requency sound waves
or cats that have excellent sense of smell in spite of poor vision and
many more.

13.2 WHAT IS AN IMAGE?
 An image can be defined as a 2 -dimensional representation of 3 -
dimensional world.
 The world around us is 3 -dimensional
 Images obtained through a camera are 2 -dimensional.
 Hence image can be defined as a two -dimensional function, f(x, y)
where x and y are spatial coordinates and the amplitude of f at any pair
of coordinates (x, y) is called the intensity or grey level of the imag e at
that point.
 When x, y and the intensity values of f are all finite discrete quantities
we say the image is a digital image.
 A digital image is composed of a finite number of elements, each of
which has a particular location and value.
 These elements are called image elements, picture elements, pels or
pixels.
 Pixel is the term used most widely to denote the elements of a digital
image.
 As we discuss earlier vision is the most advanced of our senses,
therefore images play the single most important role in human
perception system.

Fig.13.2.1
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 In 2-dimension image we lose one dimension.
 Depth information is lost in this moving process from 3 -dimension
object to 2-dimension image as shown in Fig. 13.2.1.
 Photographs on identity card, all family pictures etc. are example of 2 -
dimensional image .

FIG 13.2.2

FIG 13.2.3 FIG 13.2.4
 In Fig. 13.2.2, to locate the pixel we need to know its position in the two
directions x and y.
 Here (x, y) are spatial coordinates and f is grey level (colour in case of
colour images) at that point.
 Therefore all the images we see are 2 –dimensional function.
 A typical image is represented as shown in Fig. 13.2.3 and Fig. 13.2.4.

13.2.1 Types of Image :
It was stated earlier images are 2-dimensinal functions. Images can be
classified as follows:
(1) Monochrome Image
(2) Grey Scale Image
(3) Colour Image

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Fig. 13.2.1.1 Fig. 13.2.1.2
(1) Monochrome Image:
 In this type of image, each pixel is stored as a single bit 0 or 1.
 These images are also called as bit mapped image or binary image.
 Here 0 represen t black and 1 represent white.
 It is also c alled as black and white image.
 In this image we have black and white pixels and no sha des of grey
as shown in Fig. 13.2.1.1 and Fig. 13.2.1.2.

(2) Grey Scale Image:
 In this type of image, each pixel is usually stored as 8 -bits (1 byte).
 Because of this each pixel can have value ranging from 0 to 255.
 Here 0 represent black and 255 represent white.
 In this image we have black, white and various shades of grey
present in image as shown in Fig. 13.2.1.3 and Fig 13.2.1.4.


Fig. 13.2.1.3 Fig. 13.2.1.4

(3) Colour Image:
 In this type of image, each pixel is usually stored as 24 -bits.
 Variety of colours can be generated by mixing the three primary
colours Red, Green and B lue in appropriate proportions.
 In colour images, each pixel is composed of RGB values and each
of colours requires 8 -bits for its representation.
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 Hence each pixel is represented by 24 -bits[R (8 bits) , G ( 8
bits),B(8 bits)].
Colour images can be easily converted to grey scale image using
the equation,
X = 0.30R +0.59G+0. 13B … (13.2.1.1)
Easier formula to achieve same result as follows,
X = R + G + B / 3 … (13.2.1.2)

13.3 DIGITAL IMAGE FILE FORMATS
 A file format is a structure which defines how information is stored in
the file and how that information is displayed on the monitor.
 There are numerous image file formats which are available.
 Some of the image formats that can be used on either Macintosh or PC
platforms are:
1. BMP (Bit Mapped Graphic Image)
2. JPEG (Joint Photographic Expert Group)
3. TIFF (Tagged Image File Formal)
4. EPS (Encapsulated Post Script)
5. GIF (Graphic Interchange Format)
6. PICT (Macintosh Supported)
7. PNG (Portable Network Graphics)
1. BMP :
 Images developed by Microsoft can save both monochromes as well as
colour images.
 All the wallpaper on computer is example of BMP images.
 When we work in paint Brush, we can save our work as BMP image.
 The quality of BMP files is very good but they occupy a lot of memory
space.

2. JPEG :
 Joint Photographic E xpert Group is the name of committee that
developed this JPEG image format which is used in compression
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 It is based on the Discrete Cosine Transform -DCT .
 JPEG images use lossy compression algorithms which result in some
loss of original data.
 The quality of JPEG images is not as good as BMP images.

3. TIFF :
 Most well -known formats were developed in 1986.
 A TIFF format is also a data compression technique for monochrome as
well as colour images.
 Images seen on the Internet sites are normally TIFF/GIF images simply
because they occupy less space.

4. EPS:
 It is file format of the post script page description language and is
device in dependent.
 This means that i mages can readily be transferred from one application
to another.
 However EPS image scan only be printed in script compatible printers.

5. GIF:
 Uses a form of Huffman coding.
 Used in Internet images
 Limited color palette with RGB values, in the range of 0 to 255.
 CMYK (Cyan, Magenta, Yellow, and Key (black)) colors are not
possible in GIF.

6. PICT:
 It is a general purpose format supported by Macintosh and used for
storing bit-mapped images.

7. PNG :
 It is lossless image format.
 To improve and replace the GIF format, PNG image format was
designed.
 The image file shows organisation of information encoded that consists
of two parts:
1. Header 2. Image data.
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Fig.13.3.1
 Header: The header file gives us the information about the kind of
image.
 Format: The header file identifies the format as binary code or ASCII
string .
 Width and Height: The width and height of the image are given in
number of pixels.
Common image types : binary images, 8 -bit greyscale images, 8-bit
colour and 24-bit colour images .
 Image data format : It specifies the order in which pixel values are
stored in the image data section. Order used is left to right and top to
bottom.
 Image data format also specifies if the RGB valu es in the image are
interlaced or not interlaced.
 Example of interlaced: R G B R G B R G B
1 pixel 2 pixel 3 pixel
 Example of not interlaced: R R R…. G G G… . B B B ….
 By interlaced we mean that the RGB values of each pixel stay together
consecutively followed by the three colour components for the next
pixel .
 If the RGB values are not interlaced, the values of one primary colour
for all pixels appear first, then the values of the next primary colour
followed by the values of the third primary colour.
 Compression type : It shows image data is compressed .
 There are a few more formats given below:
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1. PGM (Portable Grey Map): Represent grey level image .
2. PBM (Portable Bit Map) : Represent binary images.
3. PPM (Portable Pixel Map) :Represent colour images.

13.4 IMAGE COMPRESSION
 In comparison to text files, images take up a huge amount of disk
space.
 The term data compression refers to the process of reducing the amount
of data required to represent a given quantity of information.
 Various amounts of data may be used to represent same amount of
information.
 This is known as data redundancy.
 Data redundancy is central issue in digital image compression.
 Basically t here are three basic redundancies:
1. Coding redundancy. 2. Interpixel redundancy. 3.Pyschovisual
redundancy.
 Data compression is achieved when one or more of these redundancies
are eliminated.
 In image compression based on methodologies, there are two types of
compression as follows:
 Lossy compression: Part of original data is discarded permanently to
reduce file size. After decompression original data cannot be retrieved
back.
Advantage: No degradation of quality.
 Lossless compression: Original data is not changed permanently
during compression. After decompression original data can be retrieved
back.
Disadvantage: Degradation of quality.

13.4.1 IMAGE COMPRESSION STANDARD -JPEG :
 JPEG stands for Joint Photographic ExpertGroup .
 Image compression is known to most users of computers in the form of
image file extensions such as the jpg file extension .
 JPEG images are compressed images which occupy very little space.
 It is based on the Discrete Cosine Transform ( DCT ).
 JPEG images use lossy compression algorithms .
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JPEG Compression Process:
 First, we convert the R, G, B color format to Y, Cb and Cr format.
 Some colors are more sensitive to human eye, thus are high -frequency
color and some of chromium compounds like Cb and Cr a re less
sensitive thus can be ignored.
 Then we reduce the size of pixels by down sampling process .
 We divide our image into 8*8 pixels and perform forward DCT (Direct
Cosine Transformation).
 Then we perform quant ization using quantum tables.
 Finally we compress our data usi ng various lossless compression
methods such as run-length enc oding, Huffman en coding etc.

Fig.13.4.1.1
13.5 IMAGE PROCESSING
 Image Processing is a subject which basically studies the images and
manipulation of images by computer.
 It involves processing the images that are obtained by camera.
 With the beginning of computers, image processing grew rapidly as a
subject.
 It was stated earlier that human eye -brain mechanism represents
ultimate imaging system.
Now let us understand this with a scenario.

What happens when we look at an object?
Let us take a simple example: when we see an explosion, we immediately
identify it as something danger. Two things have happened here:
1. The scene has been captured by the eye.
2. The brain processed the scene and gave out a warning signal.

This is known as Image Processing.
 In above example the scene is captured by the eye and it sends the
signal to the brain.
 Brains processed this signal and some meaningful information is
generated.
 When working with computer, images captured from camera are fed
into the system where algorithm is written to process these images.
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 Hence in image processing system, camera replaces the human eye and
computer replaces brain which does the processing.
 The field of digital image processing refers to processing digital images
by means of a digital computer.

Fig.13.5.1
 Therefore w e start processing the images the day we are born and
continue t ill we die.
 Hence image processing is integral part of us.
 An another important trait that is common to all human being is to
store the information, analyse it, discuss it with others and try to make
it better.
13.5.1 DIGITAL IMAGE ENHANCEMENT :
 Image enhancement is the next step after image acquisition in
fundamental image processing.
 There are many improvisation methods to manipulate an images
and the result is more suitable than the original image .
 Here image is enhanced.
 The desired result can vary from person to person.
 In image enhancement no extra information is added to the
original image.
 It simply improves the subjective quality of an image by
working wit h the existing data of an image.
 Image enhancement can be carried in two domain s:
1. Spatial domain
2. Frequency domain
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Fig: 13.5.1.1
1. Spatial Domain:
 Spatial domain means working in the given space.
 It implies working with the pixels values of an image .
 Assume f(x, y) is original image. Here f is grey value and (x, y) are
image coordinate.
 For 1 byte image f values range from 0 -255, where 0 represent black,
255 represent white and all intermediate values represent shades of
grey.
 The modified image can be expressed as,
g (x, y) = T [f(x, y)]
Where g (x, y) is modified image .
f (x, y) is original image .
T is the transformation applied on original image.
 Spatial domain enhancement can be carried out in two ways:
Point processing and Neighbourhood processing.

2. Frequency Domain:
 Frequency domain means working with the frequencies value of
an image.
 Frequency domain techniques are the study of working in the
Fourier domain.
 Example: 1D -Fourier transforms , 2D-Fourier transform, 1D-DFT
and 2D -DFT etc.

13.5.2 CONTRAST STRETCHING :
 We obtain low contrast images because many reasons such as
poor illumination , lack of dynamic range in the imaging sensor
or wrong s etting of a lens aperture during image acquisition.
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 Contrast stretching expands the range of intensity levels in an
image .
 This means that contras t stretching increases dark portion to
darker and bright portion to brighter.
 Contrast stretching is also used to increase the dynamic range of
the modified image.
 Fig 13.5.2.1 shows transformation used to achieve contrast
stretching.


Fig. 13.5.2.1
 In figure dotted line indicate identity transformation and solid
line indicate contrast stretching transformation.
 Here the dark grey levels are made darker by assigning slope
value less than one and bright grey levels are made brighter by
assigning slope value greater than one.
 The equation of the contrast stretching algorithm is given as:
g ( x, y )= l.f ( x, y ) for 0 <= r < a or
g(x, y) = m.( r – a ) + v for a <= r < b or
g( x, y) = n.( r – b ) + w for b <= r < L -1
where g ( x, y ) is modified image.
f ( x, y ) is original image.
l, m, n are slopes.
 It is clear from figure that l and n slope values are less than one
while m slope is greater than one.
 Fig 13.5.2.2 shows original image and Fig 13.5.2.3 shows
contrast stretched image .

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Fig. 13.5.2.2 Fig. 13.5.2.3

13.5.3 HISTOGRAM :
 Histogram of images provides a global description of the
appearance of an image.
 The information obtained from histograms is enormous.
 Histogram of an image represents the relative frequency of
occurrence of the various grey levels in an image.
 Histogram of an i mage can be plotted in two ways.
 In the first method, the x -axis has the grey levels and the y -axis has the
numbe r of pixels in each grey level.
 In the second method, the x-axis represents the grey levels , while the y-
axis represents the probability of the occurrence of that grey level.


Method 1

Method 2



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 In method 2,
P (rk) = n k / n
rkkth grey level
nk Number of pixel in the k th grey level
n Total number of pixel in an image

13.5.3.1 HISTOGRAM EQUILIZATION :
 There are many applications where we need flat histogram.
 Histogram equalization gives us flat histogram.
 A perfect image is one which has equal number of pixels in all its grey
level, which is achieved by ideal histogram equalization.
 This technique carries the transformations that transform bad histogram
to a flat histogram.
 It also increases the dynamic range of a given image along with equal
distribution of pixels in its grey levels.
 The transformation we are looking for must satisfy the following two
conditions:
1. T( r ) must be single valued and monotonically increasing in the
interval 0 <= r <= 1.
2. 0 <= T( r ) <= 1 for 0 <= r <= 1 i.e. 0 <= s <= 1 for 0 <= r <= 1 .
Here range of r is [0 , 1].This is called normalized range.


Original Image Histogram

Equalized Image Equalized Histogram
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13.5.4 FILTERING :
 As we studied in image enhancement topic, neighbourhood
processing is one of the important technique in spatial domain.
 In neighbourhood processing we consider pixel as well its
immediate neighbours to enhance the image.


Fig. 13.5.4.1 Fig. 13.5.4.2

 Fig.13.5.4.2 shown is called filter, mask, window or a
template.
 Mask can be 3 x 3, 5 x 5, 7 x 7 and so on.
 To process image we place mask on image, multiply each
component of mask with corresponding value of image, add
them up and place the calculated value at center.
 Consider f(x, y) is the original image and g(x, y) is modified
image then,

g(x,y)=w1*f(x -1,y-1)+w2*f(x -1,y)+w3*f(x -1,y+1)+w4*f(x,y -1)
+w5*f(x, y)+w6*f(x,y+1)+w7*f(x+1,y -1)+w8*f(x+1,y)
+w9*f(x+1,y+1)
 Once g(x, y) is calculated we shift t he mask by one step
towards the right to the next pixel.
 One of the important operations of neighbourhood
processing is image filtering.
 In filtering we remove the noise from the image.
 In most images, the background is considered to be low
frequency regio ns and edges are considered to be high
frequency region.

13.5.4.1 SMOOTHING :
 Smoothing is also called as low pass filtering.
 Smoothing implies removing or blurring the edges.
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 Smoothing is used to remove noise present in the image.
 Noise is normally high frequency component and low pass
filtering removes these high frequency components.
 Here we are going to study smoothing in two ways: Low Pass
Averaging Filter and Median Filter .

13.5.4.1.1 LOW PASS AVERAGE FILTER
 This filter is us ed to eliminate Gaussian noise from the image.
 Frequency response of low pass averaging filter along with
spatial response is shown in figure.
 We generate the mask which gives us low pass filtering
operation.
 The coefficient use in mask should be positive in case of low
pass averaging filter.
.
Fig.13.5.4.1.1.1
 Let us take the 3 x 3 average masks shown in Fig. 13.5.4.1.1.1 on
the pseudo image given below to understand working of low pass
averaging filter.
 Consider the size of image is 8 x 8.
 It is clear that 8 x 8 image have single edge between 10 and 50.
 Place the 3 x 3 average masks on this image.
 Start from top left hand corner.
 Multiply each component of image with the corresponding value
of mask.
 As all values of image inside mask are 10, we get aver age as 10
and center pixel remains 10.
 Therefore according to equation of neighbourhood processing,

g(x, y)=1*10+1*10 +1*10+1*10+1*10+1*10+1*10+1*10+ 1*10/9
=10+10+10+10+10+10+10+10+10/9
=90/9
=10
Next shift the mask toward right till we reach end of line, perform
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same step of calculation as explain above and then move it
downwards.
 Some of calculation of mask position is shown below.




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 Once we g et final result you can observe that the low frequency
region has remained unchanged. Hence we can conclude that
sharp edge has become blur.
 The sharp edge between 10 and 50 has become blurred.
 We can see from grey level 10 it changes to 23.3(after rounding
off its 23), from 23 it changes to 36.6(after rounding off its 36)
and finally from 36 it changes to 50.

13.5.4.1.2 LOW PASS MEDIAN FILTER :
 We have seen low pass averaging filter blurs the edges by
eliminating the noise.
 Bigger the averaging mask more is blurring.
 Also there are chances image can contain salt and pepper noise.
 If we work with low pass averaging filter to remove the same it
will blur noise but at same time it will also ruin the edges.
 Therefore we need to work with non -linear filter.
 Such non -linear filter is called as Median filter.
 Low pass median filter removes the salt and pepper noise from
the image
 They are also called as order statistic filters because their
response is based on ranking of the pixels contained within the
mask.
 Consider using the mask similar to av eraging filter except that
the mask will contain no values.
 The step to perform median filtering is as follows:
1. Assume 3 x 3 empty mask.
2. Place the empty mask at top left hand corner of given pseudo
image.
3. Arrange all nine pixels inside mask in ascending or
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descending order.
4. Choose the media n value from these nine pixels and place at
center.
5. Move the mask in similar fashion as done in averaging filter.
 Consider the 8 x 8 pseudo image with salt and pepper noise given
below.
 We need to remove noise without dis turbing the edges.
 Place the empty mask at top left hand corner.
 Arrange the nine pixels of image inside mask either in ascending
or descending order as given below.
10 10 10 10 10 10 10 10 250
 The median is value placed at fifth position. Therefore this value
is placed at the center.
 Continue this procedure until all salt and pepper noise get
replaced by median value.



 Hence we get the final result as salt and pepper noise got eliminated
without distortion of edges.



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13.6 S UMMARY
 Image is 2 dimensional representation of 3 dimensional World.
 We can store different types of images such as binary image, grey scale
image, 8-bits colour image, and 2 4 bits colour image.
 Image processing is a study in which images are recorded by camera
and are processed by the computer.
 Various file formats are used such as jpeg, gif, bmp etc.
 Image data can be compressed and good image compression standard
like jpeg is available.
 There are image enhancement methods like contrast stretching,
histogram equalization and filtering which subjectively improvised and
enhance the original image into more suitable one.

13.7 REFERENCES
• Digital Image Processing (Using Matlab Codes) by Dhananjay K.
Theckedath.

13.8 BIBLIOGRAPHY
1. Digital Image Processing Fourth Edition Rafael C. Gonzalez and
Richards E. Woods.

13.9 EXERCISE
1. Define Images. State and explain types of i mages.
2. Write a short no te on histogram equalization.
3. Explain low pass averaging filter and low pass median filter .
4. Write a short note on image enhancement .
5. List and explain image file formats.
6. Explain image compression jpeg standards.


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