Computer Graphics

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@ Ashek Mahmud Khan; Dept. of CSE (JUST); 01725-402592

01725-402592

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Computer Graphics (Chapter:01)

Q 1. Define Computer Graphics.

Computer graphics: Computer graphics may be defined as a pictorial representation or

graphical representation of objects in a computer.

Computer graphics are pictures and movies created using computers usually referring to

image data created by a computer specifically with help from specialized graphic hardware

and software.

Q. 2013-1(b)/2012-1(a): Mention the Application of Computer Graphics and explain.

(i) Computer Aided Design (CAD)

(ii) Presentation Graphics

(iii) Computer Art

(iv) Entertainment

(v) Education & Training

(vi) Visualization

(vii) Image Processing

(viii) Graphical user Interfaces

(i) Computer Aided Design (CAD): A major use of computer graphics is in design

processes, particularly for engineering and architectural systems, but almost all products are

now computer designed. Generally referred to as CAD, computer-aided design methods are

now routinely used in the design of buildings, automobiles, aircraft, watercraft, spacecraft,

computers, textiles, and many, many other products.

(ii) Presentation Graphics: Another major application area is presentation graphics, used to

produce illustrations for reports or to generate 35-mm slides or transparencies for use with

projectors. Presentation graphics is commonly used to summarize financial, statistical,

mathematical, scientific, and economic data for research reports, managerial reports, and

consumer information bulletins.

(iii) Computer Art: Computer graphics methods are widely used in both fine art and

commercial art applications.

(iv) Entertainment: Computer graphics methods are now commonly used in making motion

pictures, music videos, and television shows.

(v) Education & Training: Computer-generated models of physical, financial, and

economic systems are often used as educational aids.

Ashek Mahmud Khan

Dept. of CSE (JUST)

01725-402592

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http://en.wikipedia.org/wiki/Image

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(vi) Visualization: Scientists, engineers, medical personnel, business analysts, and others

often need to analyze large amounts of information or to study the behavior of certain

processes.

(vii) Image Processing: In computer graphics, a computer is used to create a picture. Image

processing, on the other hand applies techniques to modify or interpret existing pictures,

such as photo graphs and TV scans.

(viii) Graphical user interface: It is common now for software packages to provide a

graphical interface. A major component of a graphical interface is a window manager that

allows a user to display multiple-window areas. Each window can contain a different process

that can contain graphical or non graphical displays.

Q. 2012-1(a): What do you mean by GUI?

GUI stands for Graphical user interface. A major component of a GUI is a window manager

that allows a user to display multiple-window areas. To make a particular window active we

simply click in that window using an interactive pointing device. Interfaces also display menus

and icons for fast selection of processing options or parameter values.

Q. 2013-1(a): Define: (i) Pixel (i) Resolution (i) Aspect ratio (i) Persistence

Pixel: Each screen point is referred to as a pixel or pel. The pixel (a word invented from

"picture element") is the basic unit of programmable color on a computer display or in a

computer image. Think of it as a logical - rather than a physical - unit. The physical size of a

pixel depends on how you've set the resolution for the display screen.

Resolution: Resolution is defined as a matrix of "pixels" per inch. Screen Pixels Per Inch. A

screen resolution of 1920x1200 means 1,920 pixels horizontally across each of 1,200 lines,

which run vertically from top to bottom.

Aspect ratio: Aspect ratio is an image projection attribute that describes the proportional

relationship between the width of an image and its height. For example, if a graphic has an

aspect ratio of 2:1, it means that the width is twice as large as the height.

Persistence: Persistence mean how long the electronic beam continue emit light after the

CRT beam is removed.

Chapter:02-(Overview of Graphics Systems)

Video Display Device: The primary output device in a graphics system is a video monitor.

The operation of most video monitors is based on the standard cathode-ray tube (CRT)

design.

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Figure 2-1 Basic design of a magnetic-deflection CRT

The basic operation of a CRT: A beam of electrons (cathode rays) emitted by an electron

gun, passes through focusing and deflection systems that direct the beam toward specified

positions on the phosphor coated screen.

The phosphor then emits a small spot of light at each position contacted by the

electron beam. This type of display is called a refresh CRT.

The primary components of an electron gun in a CRT are the heated metal cathode

and a control grid.

The focusing system in a CRT is needed to force the electron beam to converge into a

small spot as it strikes the phosphor.

Magnetic deflection has two pairs of coils are used, with the coils in each pair

mounted on opposite sides of the neck of the CRT envelope.

Figure:2-2 Operation of an electron gun with an accelerating anode.

Figure: 2-3 Electrostatic deflection of the electron beam in a CRT.

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Electrostatic deflection has two pairs of parallel plates are mounted inside the CRT

envelope.

Resolution: The maximum number of points that can be displayed without overlap

on a CRT is referred to as the resolution.

Q. 2013-1(c): Describe the Raster-scan display system.

Raster-Scan systems: The operation of the display device is controlled by a special-purpose

processor called video controller or display controller.

Fig: Architecture of a simple raster graphics system

Frame buffer can be anywhere in the system memory & video controller access the frame

buffer to refresh the screen

Other processors such as coprocessors & accelerators implement various graphics

operations.

Fig: Architecture of a raster system with a fixed portion of the system memory reserved for the frame

buffer.

Video controller resets the register to the first pixel position on top scan line & the refresh

process starts over.

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Fig: Basic video-controller refresh operations.

Refresh operations

X, Y register used to indicate pixel position

Fix Y register and increment X register to generate scan line

Double buffering

Pixel value can be loaded in buffer while

Provide a fast mechanism for real-time animation generation

Fig: Architecture of Raster scan graphics system with a display processor

Advantages of raster scan:

Decrease memory costs

High degree realism is achieved in picture - advanced shading & hidden surface

technique.

Computer monitors and TVs use this method

Disadvantages of raster scan:

Lines produced are zigzag as the plotted values are discrete.

Resolution is low.

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Random scan System: Random scan monitors draw a picture one line at a time and for this

reason are also referred to as vector displays.

Fig: Architecture of a simple random scan system.

The organization of a simple random-scan (vector) system is shown in Fig. An application

program is input and stored in the system memory along with a graphics package. Graphics

commands in the application program are translated by the graphics package into a display

file stored in the system memory.

Advantages of random scan:

Very high resolution, limited only by monitor.

Easy animation, just draw at different position.

Requires little memory.

Disadvantages of random scan:

Requires “intelligent electron beam, i.e., processor controlled.

Limited screen density before have flicker, can’t draw a complex image.

Limited color capability

Q. 2012-1(c): Distinguish between Raster Scan CRT and Random Scan CRT

Random Scan System (Vector) Raster Scan System

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Cannot draw realistic shaded scenes. Used in Systems to Display Realistic

Images.

Store - Line drawing instruction Store – Value related to pixels (Intensity

value)

Higher Resolution No Support – High Resolution

Smooth line drawings Capable of Producing Curves Better

Mathematical functions are used to draw

an image.

Screen points/pixels are used to draw an

image.

Cost is more. Cost is low.

Color CRT Monitors: A CRT monitor displays color pictures by using a combination of

phosphors that emit different -colored light. By combining the emitted light from the

different phosphors, a range of colors can be generated. The two basic techniques for

producing color displays with a CRT are

1. The Beam-Penetration method.

2. The Shadow-Mask method.

The Beam-Penetration method: The beam-penetration method for displaying color

pictures has been used with random-scan monitors. Two layers of phosphor, usually RED

and GREEN, are coated onto the inside of the CRT screen, and the displayed color depends

on how far the electron beam penetrates into the phosphor layers.

A beam of slow electrons excites only the outer RED layer.

A beam of very fast electrons penetrates through the RED layer and excites the inner

GREEN layer.

At intermediate beam speeds, combinations of red and green light are emitted to show two

additional colors, ORANGE and YELLOW.

Advantage:

Beam penetration has been an inexpensive way to produce color in random -scan monitors,

Disadvantage:

Only four colors are possible, and the quality of pictures is not as good as with other

methods.

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Q. 2013-1(d): What is shadow mask?

Shadow mask: The shadow mask is one of the technologies used to manufacture cathode

ray tube (CRT) televisions and computer displays that produce color images. A shadow-

mask CRT has three phosphor color dots at each pixel position.

One phosphor dot emits a RED Light, another emits a GREEN light, and the third emits a

BLUE light.

Flat-panel display: Although most graphics monitors are still constructed with CRTs, other

technologies are emerging that may soon replace CRT monitors.

The term flat-panel display refers to a class of video devices that have

1. Reduced volume

2. Weight

3. Power requirements compared to a CRT.

Current uses for flat-panel displays include small TV monitors, calculators, pocket video

games, laptop computers. Flat-panel displays into two categories:

1. Emissive displays

2. Non-Emissive displays.

Emissive displays (or emitters): These devices that convert electrical energy into light.

Examples: 1. Plasma panels. 3. Light-Emitting Diodes (LED).

Non-emissive is plays (or non-emitters): These device use optical effects to convert

sunlight or light from some other source into graphics patterns. Example: Liquid-Crystal

Device (LCD).

Plasma panel display: Plasma panels also called gas-discharge displays.

These are constructed by filling the region between two glass plates with a mixture of gases

that usually includes neon.

Advantage of LCD: (i) Low power Consumption. (ii) Low cost. (iii) Small size.

Disadvantage of LCD: (i) LCD have less color capability. (ii) Resolution is not as good as

that of a CRT. (iii) LCD’s do not emit light.

LED Advantages:

1) Better black levels (most of the time).

2) Higher contrast ratios (most of the time).

3) 40% less energy use than a florescent tube back lit LCD TV.

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5) LED will not lose color accuracy with age as florescent tubes do.

6) Extremely thin profile.

LCD Disadvantages

Aspect Ratio The aspect ratio and resolution are fixed.

Contrast Lower contrast than CRTs due to a poor black-level.

Resolution Works best at the native resolution.

Viewing Angle Restricted viewing angles

Input device: An input device is any device that provides input to a computer. Examples of

input devices include keyboards, mouse, scanners, digital cameras and joysticks.

Mouse: A device that controls the movement of the cursor or pointer on a display screen. A

mouse is a small object you can roll along a hard, flat surface. Its name is derived from its

shape, which looks a bit like a mouse.

Keyboard: External input device used to type data into some sort of computer system

whether it be a mobile device, a personal computer, or another electronic machine. A

keyboard usually includes alphabetic, numerical, and common symbols used in everyday

transcription.

There are many types of computer keyboards. For instance, multimedia computer keyboard,

membrane computer keyboard, slim computer keyboard, capacitive key switches etc.

Digitizer: Digitizer converts analog or physical input into digital image. This makes them

related to both scanners and mice, although current digitizer serve completely different role.

Scanners: A computer scanner optically scans an object such as document and data convert

the information into a digital image. The two commonly used types are: (i) Flatbed scanner

(ii) Hand-held.scanner.

Q. 2013-2(c): What is Image Scanner?

Image Scanner: In computing, an image scanner is a device used to transfer images or text

into a computer. There are special models for scanning photo negatives, or to scan books. In

the computer, the signal from the scanner is transferred to a digital image.

Joystick: A joystick is an input device consisting of a stick that pivots on a base and reports

its angle or direction to the device it is controlling. A joystick, also known as the control

column.

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Light pens: A light pen is a computer input device in the form of a light-sensitive wand

used in conjunction with a computer's CRT display. Light pens have the advantage of

'drawing' directly onto the screen, but this can become uncomfortable, and they are not as

accurate as digitizing tablets.

What are the advantages of laser printers?

1] High speed, precision and economy.

2] Cheap to maintain.

3] Quality printers.

4] Lasts for longer time.

5] Toner power is very cheap.

Chapter: 03

Points and lines: Point plotting is accomplished by converting a single coordinate position

furnished by an application program into appropriate operations for the output device in use.

With a CRT monitor, for example, the electron beam is turned on to illuminate the screen

phosphor at the selected location.

Line drawing is accomplished by calculating intermediate positions along the line path

between two specified endpoint positions. An output device is then directed to fill in these

positions between the endpoints. For analog devices, such as a vector pen plotter or a

random-scan display, a straight line can be drawn smoothly from one endpoint to the other.

Line-drawing Algorithms:

The Cartesian slope-intercept equation for a straight line:

y= m. x +b

Where m is the line slop and b is y intercept.

For line segment starting in (x1,y1) and ending in (x2,y2), the slop is:

m= (y1-y2)/(x1-x2)

Fig: Pixel positions referenced by scan-line

number and column number.

Stair step effect (jaggies) produced when

a line is generated as a series

of pixel positions.

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b= y1- m.x1

For any given x interval x, we can compute the corresponding y interval y:

y= m.x

Or x interval x from a given y:

x= y/m

|m| < 1, x can be set proportional to a small horizontal deflection voltage

|m| >1, y can be set proportional to a small vertical deflection voltage with the

corresponding horizontal deflection voltage set proportional to x.

m = 1, x = y and the horizontal and vertical deflections voltages are equal.

DDA (digital differential analyze) Algorithm: A scan-conversion line algorithm based on

calculating either y or x using line points calculating equations. In each case, choosing

the sample axis depends on the slop value.

Sample at unit x interval (x=1) and compute each successive y value as :

y k+1= yk+ m

Subscript k takes integer values starting from 1, for the first point, and increases by 1

until the final endpoint is reached. Since m can be any real number between 0 and 1,

the calculated y values must be rounded to the nearest integer.

For lines with a positive slope greater than 1, we reverse the roles of x and y. That is, we

sample at unit y intervals (y = 1) and calculate each succeeding x value as

x k+1= xk + (1/m)

If the absolute slop is less than 1, set x=-1 and

y k+1= yk - m

or, If the absolute slop is greater than 1, set y=-1 and

x k+1= xk - (1/m)

Bresenham’s Line Algorithm:

Scan converts lines using only incremental integer calculations.

Can be adapted to display circles and other curves.

When sampling at unit x intervals, we need to decide which of two possible pixel

positions is closer to the line path at each sample step by using a decision parameter.

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The decision parameter (pk) is an integer number that is proportional to the difference

between the separations of the two pixel positions from the actual line path.

Depending on the slop sign and value the decision parameter determine which pixel

coordinates would be taken in the next step.

Algorithm:

At pixel (xk,yk) we need to decide whether (xk+1,yk) or (xk+1,yk+1) would be plotted in

column xk+1.

d1 and d2 are the vertical pixel separations from the mathematical line path.

y coordinate on the mathematical line at pixel xk+1 is calculated as:

y= (x+1.m)+b

d1= y-yk

d2= yk+1-y

We accomplish this by substituting m = y/x, where y and x are the vertical and

horizontal separations of the endpoint positions, and defining:

Pk =x(d1-d2)

=2y.xk-2x. yk+c

If the pixel at yk is closer to the line path than the pixel at yk+l (that is, d1 < d2), then decision

parameter pk is negative and we plot the lower pixel.

Pk+1 =2y.xk+1-2x. yk+1+c

Subtracting Eq. from the preceding equation, we have

Pk+1-Pk =2y(xk+1- xk )- 2x(yk+1-yk)

But xk+1, = xk + 1, so that

pk+1 = pk+2y-2x(yk+1-yk)

and p0=2y-x

Bresenham's Line-Drawing Algorithm for |m|< 1

1. Input the two line endpoints and store the left endpoint in (x0, y0).

2. Load (x0, y0) into the frame buffer; that is, plot the first point.

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3. Calculate constants x, y, 2y, and 2y - 2x, and obtain the starting value for the

decision parameter as

P0 = 2y - X

4. At each xk along the line, starting at k = 0, perform the following test:

If Pk < 0, the next point to plot is (xk +1, yk) and

Pk+1 = Pk + 2y

Otherwise, the next point to plot is (xk +1, yk +1) and

pk+1, = pk + 2y - 2x,

5. Repeat step 4 x times.

Example 3-1: Bresenham Line Drawing

To illustrate the algorithm, we digitize the line with endpoints (20, 10) and (30,18). This line

has a slope of 0.8, with

x= x2-x1=30-20=10, y=y2-y1=18-10=8

The initial decision parameter has the value

P0 = 2y - X= 2×8-10=16-10=6

and the increments for calculating successive decision parameters are

2y=2×8=16, 2y - 2x=16-20= -4

We plot the initial point (x0, y0) = (20, 10), and determine successive pixel positions along

the line path from the decision parameter as-

A plot of the pixels generated along this line path is shown in Fig

Q. 2013-3(b): Explain the Mid-point circle Algorithm.

Samples at unit intervals and uses decision parameter to determine the closest pixel

position to the specified circle path at each step.

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Along the circle section from x = 0 to x = y in the first quadrant, the slope of the curve

varies from 0 to -1. Therefore, we can take unit steps in the positive x direction over

this octant and use a decision parameter to determine which of the two possible y

positions is closer to the circle path at each step. Positions in the other seven octants

are then obtained by symmetry.

To apply mid-point method, we define a circle function:

fcircle (x,y) = x2+y

2-r

2

and determine the nearest y position from pk:

Successive decision parameters are obtained using incremental calculations. We obtain a

recursive expression for the next decision parameter by evaluating the circle function at

sampling position xk+1 + 1 = xk+ 2:

Where yk+1 is either yk or yk-1, depending on the sign of pk.

Q. 2013-3(c)/Example 3-2: Drive successive decision values and positions along the

circle path using the midpoint method. [where r= 10 & first quadrant from x=0 to x=y]

or

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Given a circle radius r = 10, we demonstrate the midpoint circle algorithm by determining

positions along the circle octant in the first quadrant from x = 0 to x = y. The initial value of

the decision parameter is

P0 = 1-r = -9

For the circle centered on the coordinate origin, the initial point is (x0, y0) = (0, l0), and

initial increment terms for calculating the decision parameters are-

2x0=0, 2y0=20

Midpoint ellipse Algorithm:

Ellipse Properties: Expressing distances d1 and d2 in terms of the focal coordinates F1 = (x1,

y1) and F2 = (x2, y2), we have:

Cartesian coordinates:

Polar coordinates:

Explain Ellipse Algorithm:

If (x0, y0)=(0, r)

2 2 2 2

1 1 2 2( ) ( ) ( ) ( ) constantx x y y x x y y

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1c c

x y

x x y y

r r

cos

sin

c x

c y

x x r

y y r

2 2 2 2 2 2( , )ellipse y x x yf x y r x r y r r

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Decision parameter:

At the boundary:

Therefore, we move out of region 1 whenever,

Assuming that we have just plotted the pixels at (xi , yi). The next position is determined by:

If p1i < 0 the midpoint is inside the ellipse yi is closer

If p1i ≥ 0 the midpoint is outside the ellipse yi – 1 is closer

At the next position [xi+1 + 1 = xi + 2]

0 if ( , ) is inside the ellipse

( , ) 0 if ( , ) is on the ellipse

0 if ( , ) is outside the ellipse

ellipse

x y

f x y x y

x y

2

2

2

2

y

x

r xdySlope

dx r y

2 21 2 2y x

dyr x r y

dx

2 22 2y xr x r y

12

2 2 2 2 2 212

1 ( 1, )

( 1) ( )

i ellipse i i

y i x i x y

p f x y

r x r y r r

11 1 1 2

2 2 2 2 2 211 2

1 ( 1, )

( 2) ( )

i ellipse i i

y i x i x y

p f x y

r x r y r r

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Or,

where yi+1 = yi or,yi+1 = yi – 1

Midpoint Ellipse Algorithm:

1. Input rx, ry, and ellipse center (xc, yc), and obtain the first point on an ellipse centered

on the origin as

(x0, y0) = (0, ry)

2. Calculate the initial parameter in region 1 as

2. At each xi position, starting at i = 0, if p1i < 0, the next point along the ellipse centered

on (0, 0) is (xi + 1, yi) and

otherwise, the next point is (xi + 1, yi – 1) and

and continue until

4. (x0, y0) is the last position calculated in region 1. Calculate the initial parameter in

region 2 as

5. At each yi position, starting at i = 0, if p2i > 0, the next point along the ellipse centered

on (0, 0) is (xi, yi – 1) and

Otherwise, the next point is (xi + 1, yi – 1) and

Use the same incremental calculations as in region 1. Continue until y = 0.

6. For both regions determine symmetry points in the other three quadrants.

7. Move each calculated pixel position (x, y) onto the elliptical path centered on (xc, yc)

and plot the coordinate values

x = x + xc , y = y + yc

Example: 3-3:

2 2 2 2 2 21 11 1 2 2

1 1 2 ( 1) ( ) ( )i i y i y x i ip p r x r r y y

2 2 210 4

1 y x y xp r r r r

2 2

1 11 1 2i i y i yp p r x r

2 2 2

1 1 11 1 2 2i i y i x i yp p r x r y r

2 22 2y xr x r y

2 2 2 2 2 210 0 02

2 ( ) ( 1)y x x yp r x r y r r

2 2

1 12 2 2i i x i xp p r y r

2 2 2

1 1 12 2 2 2i i y i x i xp p r x r y r

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rx = 8 , ry = 6

2ry2x = 0 (with increment 2ry

2 = 72)

2rx2y = 2rx

2ry (with increment -2rx

2 = -128)

Region 1: (x0, y0) = (0, 6)

i pi xi+1, yi+1 2ry2xi+1 2rx

2yi+1

0 -332 (1, 6) 72 768

1 -224 (2, 6) 144 768

2 -44 (3, 6) 216 768

3 208 (4, 5) 288 640

4 -108 (5, 5) 360 640

5 288 (6, 4) 432 512

6 244 (7, 3) 504 384

Move out of region 1 since 2ry2x > 2rx

2y

Region 2: (x0, y0) = (7, 3) (Last position in region 1)

i pi xi+1, yi+1 2ry2xi+1 2rx

2yi+1

0 -151 (8, 2) 576 256

1 233 (8, 1) 576 128

2 745 (8, 0) - -

Stop at y = 0

2 2 210 4

1 332y x y xp r r r r

10 2

2 (7 ,2) 151ellipsep f

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Fig: Positions along an elliptical path centered on the origin with rx= 8 and ry= 6 using the midpoint

algorithm to calculate pixel addresses in the first quadrant.

Chapter: 05

Basic transformations: Basic transformations such as translation, rotation, and scaling are

included in most graphics packages. Some packages provide a few additional

transformations that are useful in certain applications. Two such transformations are

reflection and shear.

2D Transformations: “Transformations are the operations applied to geometrical

description of an object to change its position, orientation, or sizes are called geometric

transformations”.

Why Transformations?

“Transformations are needed to manipulate the initially created object and to display the

modified object without having to redraw it.”

Q.2013-5(b): What do you mean by Translation and Rotation in geometric

transformations?

Translation: A translation moves all points in an object along the same straight-line path to

new positions.

We can write the components:

p'x = px + tx

p'y = py + ty

or in matrix form:

P' = P + T

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Rotation: A two-dimensional rotation is applied to an object by repositioning it along a

circular path in the xy plane. To generate a rotation, we specify a rotation angle 0 and the

position (x1,yl) of the rotation point (or pivot point) about which the object is to be rotated

(Fig. 5-3).

Figure 5-3 Rotation of an object through angle about the pivot point (x1, y1).

We can write the components:

p'x = px cos – py sin

p'y = px sin + py cos

or in matrix form:

P' = R • P

can be clockwise (-ve) or counterclockwise (+ve as our example).

Rotation matrix:

Scaling: Scaling changes the size of an object and involves two scale factors, Sx and Sy for

the x- and y-coordinates respectively.

• Scales are about the origin.

• We can write the components:

p'x = sx • px

p'y = sy • py

or in matrix form:

P' = S • P

Scale matrix as:

cossin

sincosR

y

x

s

sS

0

0

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Q.2013-6(d):What do you know about Composite Transformations?

Composite Transformations: With the matrix representations of the previous section, we

can set up a matrix for any sequence of transformations as a composite transformation

matrix by calculating the matrix product of the individual transformations.

Translations: If two successive translation vectors (tx1,ty1) and (tx2,ty2) are applied to a

coordinate position P, the final transformed location P’ is calculated as:-

P’=T(tx2,ty2) . {T(tx1,ty1) .P}

={T(tx2,ty2) . T(tx1,ty1)} .P

Where P and P’ are represented as homogeneous-coordinate column vectors. We can verify

this result by calculating the matrix product for the two associative groupings. Also, the

composite transformation matrix for this sequence of transformations is: -

Or, T(tx2,ty2) . T(tx1,ty1) = T(tx1+tx2, ty1+ty2)

Which demonstrate that two successive translations are additive.

Rotations: Two successive rotations applied to point P produce the transformed position: -

P’= R(Ө2) . {R(Ө1 ) . P}

= {R(Ө2) . R(Ө1)} . P

By multiplication the two rotation matrices, we can verify that two successive rotations are

additive:

R(Ө2) . R(Ө1) = R (Ө1+ Ө2)

So that the final rotated coordinates can be calculated with the composite rotation matrix as:-

P’ = R(Ө1+ Ө2) . P

Scaling: Concatenating transformation matrices for two successive scaling operations

produce the following composite scaling matrix: -

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Or, S(Sx2, Sy2 ) . S(Sx1, Sy1) = S (Sx1 . Sx2, Sy1 .Sy2 )

The resulting matrix in this case indicates that successive scaling operations are

multiplicative.

General pivot point rotation: Translate the object so that pivot-position is moved to the

coordinate origin. Rotate the object about the coordinate origin.

General fixed point scaling: Translate object so that the fixed point coincides with the

coordinate origin. Scale the object with respect to the coordinate origin.

Reflection: Reflection is a transformation that produces a mirror image of an object. It is

obtained by rotating the object by 180 deg about the reflection axis

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Figure: Reflection of an object about the x axis.

Figure: Reflection of an object about the y axis.

Shear: A transformation that distorts the shape of an object such that the transformed shape

appears as if the object were composed of internal layers that had been caused to slide over

each other is called shear. Two types of shearing transformations are there about X values

and about Y values

Shearing transformations in three-dimensions alter two of the three coordinate values

proportionally to the value of the third coordinate.

Affine Transformation: A coordinate transformation of the form

x' =axxx+axyy+bx, y' =ayxx+ayyy+by

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is called a two-dimensional affine transformation. Each of the transformed coordinates x' and

y' is a linear function of the original coordinates x and y, and parameters aij and bk are

constants determined by the transformation type. Affine transformations have the general

properties that parallel lines are transformed into parallel lines and finite points map to finite

points.

Translation, rotation, scaling, reflection, and shear are examples of two-dimensional affine

transformations.

Chapter: 06

Clipping: Identify those portions of a picture that are either inside or outside of a specified

region of space. The region against which an object is to clipped is called a clip window.

Viewport clipping: It can reduce calculations by allowing concatenation of viewing and

geometric transformation matrices.

In the following sections, we consider algorithms for clipping the following primitive types-

Point Clipping

Line Clipping (straight-line segments)

Area Clipping (polygons)

Curve Clipping

Text Clipping

Point Clipping: Assuming that the clip window is a rectangle in standard position, we save

a point P = (x, y) for display if the following inequalities are satisfied:

xwmin ≤ x ≤ xwmax

ywmin ≤ y ≤ ywmax

Where the edges of the clip window (xwmin, xwmax, ywmin, ywmax) can be either the world-

coordinate window boundaries or viewport boundaries. If any one of these four inequalities

is not satisfied, the point is clipped (not saved for display)

Line Clipping: Possible relationships between line positions and a standard rectangular

clipping region

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• Possible relation ships

– Completely inside the clipping window

– Completely outside the window

– Partially inside the window

• Parametric representation of a line

x = x1+ u (x2- x1)

y = y1+ u (y2- y1)

• The value of u for an intersection with a rectangle boundary edge

– Outside the range 0 to 1

– Within the range from 0 to 1

Polygons clipping: To clip polygons, we need to modify the line-clipping procedures

discussed in the previous section. A polygon boundary processed with a line clipper may be

displayed as a series of unconnected line segments, depending on the orientation of the

polygon to the clipping window. The following example illustrates a simple case of polygon

clipping.

Curve Clipping: Use bounding rectangle to test for overlap with a rectangular clip window. Curve clipping procedures will involve non-linear equations (so requires more processing

than for objects with linear boundaries).

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Text Clipping: There are several techniques that can be used to provide text clipping in a

graphics package. The clipping technique used will depend on the methods used to generate

characters and the requirements of a particular application. Methods for processing

character strings relative to a window boundary

• All-or-none string clipping strategy

• All or none character clipping strategy

• Clip the components of individual characters

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Chapter: 10

Q. 2012-8(i): Write the short note of following terms-

Bézier curve: A Bézier curve is a parametric curve frequently used in computer graphics

and related fields.

In general, a Bezier curve section can be fitted to any number of control points. The number

of control points to be approximated and their relative position determine the degree of the

Bézier polynomial.

Clip the components of

individual characters

Treat characters same as

lines

If individual char overlaps

a clip window boundary,

clip off the parts of the

character that are outside

the window

http://en.wikipedia.org/wiki/Parametric_curve

http://en.wikipedia.org/wiki/Computer_graphics

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Fig: Cubic Bézier curve

Suppose we are given n + 1 control-point positions: pk = (xk, yk, zk), with k varying from 0 to

n. These coordinate points can be blended to produce the following position vector P(u),

which describes the path of an approximating Bezier polynomial function between p0, and

pn.

P(u) = BEZ k,n(u),

The Bezier blending function BEZ k,n(u), are the Bernstein polynomials:

BEZ k,n(u) = C(n,k) uk (1-u)

n-k

Where the C(n,k) are the binomial coefficient:

C(n,k) =

Equivalently, we can define Bezier blending functions with the recursive calculation

BEZ k,n(u) = (1-u) BEZ k,n-1(u) + u BEZ k-1,n-1(u) , n>k 1

with BEZ k,k = uk, and BEZ0,k = (1- u)

k. Vector equation represents a set of three parametric

equations for the individual curve coordinates.

x(u) = BEZ k,n(u),

y(u) = BEZ k,n(u),

z(u) = BEZ k,n(u),

0 u

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Properties of the Bézier curves:

Interpolation: A Bézier curve always interpolates the end control points.

P(0)=P0

P(1)=Pn

Tangency: The end point tangent vectors are parallel to P1- P0 and Pn- Pn-1

Convex hull property: The curve is contained in the convex hull of its defining control

points.

p (0) = n(n-1)[(p2 – p1) – (p1 – p0) ]

p (1) = n(n-1)[(pn-2 – pn-1) – (pn-1 – pn)]

Variation diminishing property: No straight line intersects a Bézier curve more times than

it intersects its control polygon.

= 1

The contrex-hull property for a Bezier curve ensures that the polynomial smoothly follows

the control points without erratic oscillations.

Bézier Surface: Two sets of orthogonal Bézier curves can be used to design an object

surface by specifying by an input mesh of control points. The parametric vector function for

the Bkzier surface is formed as the Cartesian product of Bezier blending functions:

P(u,v) = BEZ j,m(v) BEZk,n(u)

with Pi,k specifying the location of the (m + 1) by (n + I) control points.

Figure 10-39 illustrates two Bezier surface plots. The control points are connected by dashed

lines, and the solid lines show curves of constant u and constant v.

Figure 10-39: Bezier surfaces constructed tor (a) m = 3, n = 3, and (b) m = 4, n = 4. Dashed

lines connect the control points.

Q. 2012-8(ii): Write the short note of following terms-

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B-Spline Curve and Surfaces: These are the most widely used class of approximating

splines. B-splines have two advantages over Bezier splines: (1) the degree of a B-spline

polynomial can be set independently of the number of control points (with certain

limitations), and (2) B-splines allow local control over the shape of a spline curve or surface

The trade-off is that B-splines are more complex than Bezier splines.

We can write a general expression for the calculation of coordinate positions along a B-

spline curve in a blending-function formulation as-

P(u) = Bk,d(u), umin u umax

Where the pk are an input set of n + 1 control points. There are several differences between

this B-spline formulation and that for Bezier splines.

Blending functions for B-spline curves are defined by the Cox-deBoor recursion formulas:

Bk,1 (u) =

Bk,d(u) = (u) + (u)

B-spline curves have the following properties:

The polynomial curve has degree d - 1 and Cd-2

continuity over the range of u.

For n + 1 control points, the curve is described with n+1 blending functions.

Each blending function Bk,d, is defined over d subintervals of the total range of u,

starting at knot value uk.

The range of parameter u is divided into n+d subintervals by the n+d+1 values

specified in the k not vector.

Fig: Local modification of a B-spline curve. Changing one of the control points in (a) produces curve (b),

which is modified only in the neighborhood of the altered control point.

1

0

if uk u < uk+1

Otherwise

, 2 d

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Example 10-1: Uniform, Quadratic B-Splines

To illustrate the calculation of B-spline blending functions for a uniform, integer knot

vector, we select parameter values d =n= 3. The knot vector must then contain n + d + 1 = 7

knot values:

{0, 1, 2, 3, 4, 5, 6}

and the range of parameter u is from 0 to 6, with n + d = 6 subintervals.

Each of the four blending functions spans d = 3 subintervals of the total range of U. Using

the recurrence relations, we obtain the first blending function as-

B0,3 (u) =

We obtain the next periodic blending function using relationship 10-57. Substituting u-1 for

u in B0,3, and shifting the starting positions up by 1:

B1,3(u) =

Similarly, the remaining two periodic function are by successively shifting B1,3 to the right.

B2,3(u) =

u2

u(2-u)+ (u-1)(3-u)

for 0 u 1

for 1 u 2

for 2 u 3 (3-u)2

for 2 u 3

for 3 u 4

(u-1)2

(u-1)(3-u)+ (u-2)(4-u)

(4-u)2

for 1 u 2

for 2 u 3

(u-2)(4-u)+ (u-3)(5-u) ,

(u-2)2 ,

for 3 u 4

for 4 u 5 (5-u)

2,

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Chapter: 11

Geometric Transformation: The object itself is moved relative to a stationary coordinate

system or background.

With respect to some 3-D coordinate system, an object Obj is considered as a set of points.

Obj = {P(x,y,z)}

If the Obj moves to a new position, the new object Obj’ is considered:

Obj’ = { P’(x’,y’,z’)}

Translation: Moving an object is called a translation. We translate an object by translating

each vertex in the object.

x’ = x + tx

y’ = y + ty

z’ = z + tz

The translating distance pair (tx, ty, tz) is called a translation vector or shift vector. We can

also write this equation in a single

Matrix using column vectors:

x’ 1 0 0 tx x

y’ = 0 1 0 ty y

z’ 0 0 1 tz z

1 0 0 0 1 1

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Rotation: To generate a rotation transformation for an object, we must designate an axis of

rotation (about which the object is to be rotated) and the amount of angular rotation. Unlike

two-dimensional applications, where all transformations are carried out in the xy plane, a

three-dimensional rotation can be specified around any line in space. The easiest rotation

axes to handle are those that are parallel to the coordinate axes. Also, we can use

combinations of coordinate axis rotations (along with appropriate translations) to specify any

general rotation.

In 2-D, a rotation is prescribed by an angle θ & a center of rotation P. But in 3-D rotations

require the prescription of an angle of rotation & an axis of rotation.

Rotation about the z axis:

R θ,K x’ = x cosθ – y sinθ

y’ = x sinθ – y cosθ

z’ = z

Rotation about the y axis:

R θ,J x’ = x cosθ + z sinθ

y’ = y

z’ = - x sinθ + z cosθ

Rotation about the x axis:

R θ,I x’ = x

y’ = y cosθ – z sinθ

z’ = y sinθ + z cosθ

& the rotation matrix corresponding is

cos θ -sin θ 0

R θ,K = sin θ cos θ 0

0 0 1

cos θ 0 sin θ

R θ,J = 0 1 0

-sin θ 0 cos θ

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1 0 0

R θ,I = 0 cos θ -sin θ

0 sin θ cos θ

Scaling:

Changing the size of an object is called Scaling. The scale factor s determines whether

the scaling is a magnification, s > 1,Or a reduction, s < 1. Scaling with respect to the origin,

where the origin remains fixed,

x’ = x . sx

Ssx,sy,sz y’ = y . sy

z’ = z . sz

The transformation equations can be written in the matrix form:

x’ sx 0 0 x

y’ = 0 sy 0 . y

z’ 0 0 sz z

Chapter: 12

Viewing pipeline: The viewing pipeline is a group of processes common from wireframe

display through to near photo-realistic image generation, and is basically concerned with

transforming objects to be displayed from specific viewpoint and removing surfaces that

cannot be seen from this viewpoint.

World co-ordinate system: In our world co-ordinate system, we need to specify a view

reference point - this will become the origin of the view co-ordinate system.

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Finally we need to specify the view-up direction, V - this will give the y-axis direction

Q. 2013-7(a): What is Projections?

Projections: Once world-coordinate description of the objects in a scene are converted to

viewing coordinates we can project the 3D objects onto 2D view-plane. Two types of

projections: (i) Parallel Projection (ii) Perspective Projection

Parallel Projections: Coordinate Positions are transformed to the view plane along parallel

lines.

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• Orthographic parallel projection: The projection is perpendicular to the view plane.

• Oblique parallel projecion: The parallel projection is not perpendicular to the view

plane.

Perspective projection: Perspective projection transforms object positions to the view plane

while converging to a center point of projection.

Perspective projection produces realistic views but does not preserve relative proportions.

Projections of distant objects are smaller than the projections of objects of the same size that

are closer to the projection plane.

Chapter: 13 (Visible-Surface Detection Methods)

Q. 2013-8(c): Explain depth-buffer method.

Depth-buffer method: A commonly used image-space approach to detecting visible

surfaces is the depth-buffer method, which compares surface depths at each pixel position on

the projection plane. This procedure is also referred to as the z-buffer method. This method

requires 2 buffers:

1) Depth buffer or z-buffer:

• To store the depth values for each (X, Y) position, as surfaces are processed.

• 0 ≤ depth ≤ 1

2) Refresh Buffer or Frame Buffer:

• To store the intensity value or Color value at each position (X, Y).

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Figure: At view-plane position (x, y), surface S, has the smallest depth from the view plane

and so is visible at that position.

Q. 2012/09-8(a): Explain the Z-buffer algorithm.

1. depthbuffer(x,y) = 0

framebuffer(x,y) = background color

2. Process each polygon one at a time

2.1. For each projected (x,y) pixel position of a polygon, calculate depth z.

2.2. If z > depthbuffer(x,y)

Compute surface color,

set depthbuffer(x,y) = z,

framebuffer(x,y) = surfacecolor(x,y)

Calculating Depth:

• We know the depth values at the vertices.

• we can calculate the depth at any other point on the surface of the polygon using the

polygon surface equation:

C

DByAxz

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• For any scan line adjacent horizontal x positions or vertical y positions differ by 1 unit.

• The depth value of the next position (x+1,y) on the scan line can be obtained using

• For adjacent scan-lines we can compute the x value using the slope of the projected

line and the previous x value.

A-buffer method: An extension of the ideas in the depth-buffer method is the A-buffer

method (at the other end of the alphabet from "z-buffer", where z represents depth). The A-

buffer method represents an antaliased, area-averaged, accumulation-buffer method

developed by Lucas film for implementation in the surface-rendering system called REYES.

A drawback of the depth-buffer method is that it can only find one visible surface at each

pixel position. Each position in the A-buffer has two fields:

Depth field - stores a positive or negative real number

Intensity field- stores surface-intensity information or a pointer value.

C

Az

C

DByxAz

)1(

C

BmAz

mxx

/z

1

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If depth is >= 0, then the surface data field stores the depth of that pixel position as before

If depth < 0 then the data filed stores a pointer to a linked list of surface data

Surface information in the A-buffer includes:

– RGB intensity components

– Opacity parameter

– Depth

– Percent of area coverage

– Surface identifier

– Other surface rendering parameters

Depth-sorting method: Using both image-space and object-space operations, the depth-

sorting method performs the following basic functions:

1. Surfaces are sorted in order of decreasing depth.

2. Surfaces are scan converted in order, starting with the surface of greatest depth.

Q. 2013-8(b): Define BSP tree and Octree.

BSP tree: A binary space-partitioning (BSP) tree is an efficient method for determining

object visibility by painting surfaces onto the screen from back to front, as in the painter's

algorithm.

BSP-Tree Method: When BSP tree is complete, process the tree from the right nodes to the

left nodes

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Octree: An octree is a tree in which each node has at most 8 children. When an octree

representation is used for the viewing volume, hidden surface elimination is accomplished by

projecting octree nodes onto the viewing surface in a front-to-back order.

Q. 2012/09-8(c): Explain Back-Face Detection technique

Back-Face Detection: The simplest thing we can do is find the faces on the backs of

polyhedra and discard them

We know from before that a point (x, y, z) is behind a polygon surface if:

Where A, B, C & D are the plane parameters for the surface

This can actually be made even easier if we organise things to suit ourselves

Now we can simply say that if the z component of the polygon’s normal is less than zero the

surface cannot be seen

0 DCzByAx

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Q. 2013-8(a): Ray-Casting Method:

Along the line of sight, can determined which objects intersect this line

Method is based on geometric-optics methods, which trace the parts of light rays

Trace the light-ray paths backward from the pixels through the scene

Effective method for scenes with curved surfaces, particularly, spheres

Ray casting is a special case of ray-tracing algorithms

Only follow a ray out from each pixel to the nearest object

Viewport: An area on a display device to which a window is mapped. defines where it is to

be displayed.

Q. 2013-4(b):

(i) Bitmap fonts: On a black and white system with one bit per pixel, the frame buffer is

commonly known as a bitmap.

(ii) Anti-aliasing: Anti-aliasing is the smoothing of the image or sound roughness caused by

aliasing . With images, approaches include adjusting pixel positions or setting pixel intensities

so that there is a more gradual transition between the color of a line and the background color.

http://whatis.techtarget.com/definition/aliasing

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