CG Complete e Book

1 PRABHU.S

PREPARED BY S.PRABHU AP/CSE KVCET

COMPUTER GRAPHICS

CS2401

STUDY MATERIAL

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

KARPAGA VINAYAGA COLLEGE OF ENGINEERING AND TECHNOLOGY

MADURANTAKAM

S.PRABHU ASSISTANT PROFESSOR

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INDEX

SN TOPIC Page

1 Syllabus 3

2 OUTPUT PRIMITIVES 4

3 3D CONCEPTS 29

4 GRAPHICS PROGRAMMING 127

5 RENDERING 159

6 FRACTALS 182

7 QUESTION BANK 212

8 ROAD MAP 254

9 DIAGRAMS 271

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CS2401 COMPUTER GRAPHICS

UNIT I 2D PRIMITIVES

output primitives – Line, Circle and Ellipse drawing algorithms - Attributes of output

primitives – Two dimensional Geometric transformation - Two dimensional viewing –

Line, Polygon, Curve and Text clipping algorithms.

UNIT II 3D CONCEPTS

Parallel and Perspective projections - Three dimensional object representation –

Polygons, Curved lines, Splines, Quadric Surfaces,- Visualization of data sets - 3D

transformations – Viewing -Visible surface identification.

UNIT III GRAPHICS PROGRAMMING

Color Models – RGB, YIQ, CMY, HSV – Animations – General Computer Animation,

Raster, Keyframe - Graphics programming using OPENGL – Basic graphics primitives –

Drawing three dimensional objects - Drawing three dimensional scenes

UNIT IV RENDERING

Introduction to Shading models – Flat and Smooth shading – Adding texture to faces –

Adding shadows of objects – Building a camera in a program – Creating shaded objects

– Rendering texture – Drawing Shadows.

UNIT V FRACTALS

Fractals and Self similarity – Peano curves – Creating image by iterated functions –

Mandelbrot sets – Julia Sets – Random Fractals – Overview of Ray Tracing –

Intersecting rays with other primitives – texture – Reflections and

Transparency – Boolean operations on Objects

TEXT BOOKS:

1. Donald Hearn, Pauline Baker, Computer Graphics – C Version, second edition,

Pearson Education,2004.

2. F.S. Hill, Computer Graphics using OPENGL, Second edition, Pearson Education,

2003.

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UNIT I

OUTPUT PRIMITIVES

� The picture can be described in several ways.

� Picture may specify by the set of pixels in raster display.

� Or we can describe the picture as a set of complex objects.

� Such as trees and terrain or furniture and wall.

Output Primitives

� Graphics programming packages provide functions to describe a scene in terms of

these basic geometric structures, referred to as output primitives.

� To group the sets of output primitives into more complex structures.

� Each output primitive is specified with input coordinate data and other information

about the way that object is to be displayed.

Simple geometric components

� Points and straight line segments are the simplest geometric components of

pictures.

Additional output primitives

� That can be used to construct a picture include

� circles and other conic sections,

� quadric surfaces,

� spline curves and surfaces,

� polygon color areas, and

� character strings.

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POINTS_LINES_INTRODUCTION

� Shapes and colors of the objects can be described internally with pixel arrays or

with sets of basic geometric structures.

� Such as

• straight line segments and

• polygon color areas.

� The scene is then displayed either by loading the pixel arrays into the frame buffer

or by scan converting the basic geometric-structure specifications into pixel

patterns.

� Typically, graphics programming packages provide functions to describe a scene

in terms of these basic geometric structures, referred to as output primitives,

� And to group sets of output primitives into more complex structures.

� Each output primitive is specified with input coordinate data and other information

about the way that object is to be displayed.


pictures.

� Additional output primitives that can be used to construct a picture include

• circles and other conic sections,

• quadric surfaces,

• spline curves and surfaces,

• polygon color areas, and

• character strings.

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POINTS

� 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.

� How the electron beam is positioned depends on the display technology.

Random-scan system or Vector System

� It stores point-plotting instructions in the display list, and coordinate values in

these instructions are converted to deflection voltages that position the electron

beam at the screen locations to be plotted during each refresh cycle.

Black-and-white raster system

� A point is plotted by setting the bit value corresponding to a specified screen

position within the frame buffer to 1.

� Then, as the electron beam sweeps across each horizontal scan line, it emits a burst

of electrons (plots a point) whenever a value of 1 is encountered in the frame

buffer.

RGB system,

� The frame buffer is loaded with the codes for the intensities that are to be

displayed at the screen pixel positions.

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LINES

� 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.

Analog Display Devices

� 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.

� Linearly varying horizontal and vertical deflection voltages are generated that are

proportional to the required changes in the x and y directions to produce the

smooth line.

Digital Display Devices

� Digital devices display a straight line segment by plotting discrete points between

the two endpoints.

� Discrete coordinate positions along the line path are calculated from the equation

of the line.

Stair step Effect (jaggies)

� For a raster video display, the line color (intensity) is then loaded into the frame

buffer at the corresponding pixel coordinates.

� Reading from the frame buffer, the video controller then "plots" the screen pixels.

� Screen locations are referenced with integer values.

� So plotted positions may only approximate actual Line positions between two

specified endpoints.

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� For example, A computed line position of (10.48,20.51), would be converted to

pixel position (10,21).

� Thus rounding of coordinate values to integers causes lines to be displayed with a

stair step appearance ("the jaggies"), as in the following figure,

How the pixel positions are referenced?

� Pixcel positions referenced by scan line number and column number.

What is getpixel ( ) function?

� Sometimes we want to be able to retrieve the current frame buffer intensity setting

for a specified location.

� We accomplish this with the low-level function

getpixel (x, y )

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LINE-DRAWING ALGORITHMS

� The Cartesian slope-intercept equation for a straight line is

y= m.x + b

� where m representing the slope of the line and

b representing the intercept.

� Given that the two endpoints of a line segment are specified at position(x1,y1) and

(x2,y2), as shown in following Fig.

� we can determine values for the slope m and y intercept b with the following

calculations:

� Algorithms for displaying straight lines are based on the line equation 1 and the

calculations given in Eqn. 2 and 3.

� For any given x interval ⌂x along a line, we can compute the corresponding y

interval ⌂y from Eqn 2

1

2

3

4

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� Similarly, we can obtain the x interval ⌂x corresponding to a specified ⌂y as

� These equations form the basis for determining deflection voltages in analog

devices.

� For lines with slope magnitudes | m | < 1, ⌂x can be set proportional to a small

horizontal deflection voltage and the corresponding vertical deflection is then set

proportional to ⌂y as calculated from Eqn 4.

� For lines whose slopes have magnitudes | m | > 1, ⌂y can be set proportional to a

small vertical deflection voltage with the corresponding horizontal deflection

voltage set proportional to ⌂x, calculated from Eqn 5.

� For lines with m = 1, ⌂x = ⌂y and the horizontal and vertical deflections voltages

are equal.

� In each case, a smooth line with slope m is generated between the specified

endpoints.

� On raster systems, lines are plotted with pixels, and step sizes in the horizontal and

vertical directions are constrained by pixel separations.

� That is, we must "sample" a line at discrete positions and determine the nearest

pixel to the line at each sampled position.

� This scan conversion process for straight lines is shown as

5

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� Above a near horizontal line with discrete sample positions along the x axis.

DDA Algorithm

� The digital differential analyzer (DDA) is a scan-conversion line algorithm based

on calculating either ∆y or ∆x.

� We sample the line at unit intervals in one coordinate and determine corresponding

integer values nearest the line path for the other coordinate.

� Consider first a line with positive slope, as shown in Fig.

� If the slope is less than or equal to 1, we sample at unit x intervals (∆x = 1) and

compute each successive y value as

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

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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

� If this processing is reversed, so that the starting endpoint is at the right, then either

we have ∆x = - 1 and ∆y = -1

� When the start endpoint is at the right (for the same slope), we set ∆x = -1.

� Similarly, when the absolute value of a negative slope is greater than 1, we use

∆y = -1.

Advantages

� The DDA algorithm is a faster method for calculating pixel positions than older

methods.

� It eliminates the multiplication, so that appropriate increments are applied in the x

or y direction to step to pixel positions along the line path.

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BRESENHAM'S LINE ALGORITHM

� An accurate and efficient raster line-generating algorithm, developed by Bresenham.

� It use only incremental integer calculations that can be adapted to display circles and

other curves.

� Figures illustrate sections of a display screen where straight line segments are to be

drawn.

� The vertical axes show-scan-line positions.

� The horizontal axes identify pixel columns.

� Sampling at unit x intervals in these examples, we need to decide which of two

possible pixel positions is closer to the line path at each sample step.

� Starting from the left endpoint shown in Fig a, we need to determine at the next

sample position whether to plot the pixel at position (11, 11) or the one at (11, 12).

� Similarly, Fig b shows a negative slope-line path starting from the left endpoint at

pixel position (50, 50).

� In this one, do we select the next pixel position as (51,51) or as (51,49).

� These questions are answered with Bresenham's line algorithm by testing the sign

of an integer parameter, whose value is proportional to the difference between the

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

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� To illustrate Bresenham's approach, we first consider the scan-conversion process

for lines with positive slope less than 1.

� Pixel positions along a line path are then determined by sampling at unit x intervals.

� Following figure demonstrates the kth

step in this process.

� Assuming we have determined that the pixel at (xk, yk) is to be displayed, we next

need to decide which pixel to plot in column xk+1.

� Our choices are the pixels at positions (xk+1,yk) and (xk+1, yk+1).

� They coordinate on the mathematical line at pixel column position xk + l is

calculated as

Then

And

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� The difference between these two separations is

� A decision parameter pk for the kth

step in the line algorithm.

� It involves only integer calculations.

� We accomplish this by substituting m = ∆y/∆x,

� where ∆y and ∆x are the vertical and horizontal separations of the endpoint

positions, and defining:

� At step k + 1, the decision parameter is evaluated from

� Subtracting from preceding equation, we have

� at the starting pixel position (xo, yo) and with m evaluated as ∆y/∆x

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ALGORITHM

1. Input the two line endpoints and store the first end point in (x0,y0).

2. Load (x0, y0) into the frame buffer, (i.e) plot the first point.

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

for the decision parameters as

P=2∆y-∆x.

4. At each x, along the line, starting at k=0 per turn the following test.

If pk>0, the point to plot is (xk+1, yk) and

Pk+1=pk+2∆y

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

Pk+1= pk+2∆y-2∆x.

5. Repeat step 4 ∆x times.

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CIRCLE-DRAWING ALGORITHMS

� A circle is defined as the set of points that are all at a given distance r from a

center position (x, y).

� This distance relationship is expressed by the Pythagorean theorem in Cartesian

coordinates as

� Another way to eliminate the unequal spacing shown in the above figure is to

calculate points along the circular boundary using polar coordinates r and Ө.

� Expressing the circle equation in parametric polar form yields the pair of equations

1

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� Computation can be reduced by considering the symmetry of circles.

� The shape of the circle is similar in each quadrant.

� We can generate the circle section in the second quadrant of the xy plane by noting

that the two circle sections are symmetric with respect to the y axis.

� And circle sections in the third and fourth quadrants can be obtained from sections

in the first and second quadrants by considering symmetry about the x axis.

MIDPOINT CIRCLE ALGORITHM

� First we can set up our algorithm to calculate pixel positions around a circle path

centered at the coordinate origin (0,0).

� Then each calculated position (x, y) is moved to its proper screen position by

adding xc to x and yc to y.

� 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 the midpoint method, we define a circle function:

� the relative position of any point ( x , y ) can be determined by checking the

2

3

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sign of the circle function:

� Thus, the circle function is the decision parameter in the midpoint algorithm, and

we can set up incremental calculations for this function as we did in the line

algorithm.

� The above figure shows the midpoint between the two candidate pixels at

Sampling position xk + 1.

� Our decision parameter is the circle function (equation 3 ) evaluated at the

midpoint between these two pixels:

4

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� 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:

� Evaluation of the terms 2xk+1 and 2yk+1 can also be done incrementally as

� The initial decision parameter is obtained by evaluating the circle function

at the start position (x0, y0) = (0, r ) :

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� If the radius r is specified as an integer, we can simply round po to

� since all increments are integers.

ALGORITHM

1. Input radius r and circle center (xc, yc) and obtain the first point on the

circumference of a circle centered on the origin as

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

2. Calculate the initial value of the decision parameters as

p0=(5/4)-r.

3. At each xk position starting at k=0 perform the following test.

If pk<0, the next point along the circle centered on (0, 0) is (xk+1, yk) and

pk+1=pk+2xk+1+1

Otherwise, the next point along the circle is (xk+1, yk-1) and

pk+1=pk+2xk+1+1-2yk+1

Where 2xk+1=2xk+2 and 2yk+1=2yk+2

4. Determine symmetry in the other seven octants.

5. Move each calculated pixel position (x.y) onto the circular path centered on (x0, y0)

and plot the coordinate values

x=x+xc, y=y+yc

6. Repeat the step 3 through 5 until x>=y.

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Example:

To plot the pixel positions in first quadrant,

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ELLIPSE-DRAWING ALGORITHMS

� An ellipse is an elongated circle.

� Therefore, elliptical curves can be generated by modifying circle-drawing

procedures to take into account the different dimensions of an ellipse along the

major and minor axes.

Properties of Ellipses

� An ellipse is defined as the set of points such that the sum of the distances from

two fixted positions (foci) is the same for all points.

� If the distances to the two foci from any point P = (x, y) on the ellipse are labeled

dl and d2, then the general equation of an ellipse can be stated as

� we can rewrite the general ellipse equation in the form

� where the coefficients A, B, C, D, E , and F are evaluated in terms of the focal

coordinates.

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Major Axes

� The major axis is the straight line segment extending from one side of the ellipse

to the other through the foci.

Minor Axes

� The minor axis spans the shorter dimension of the ellipse, bisecting the major axis

at the halfway position (ellipse center) between the two foci.

Polar coordinate

� Using polar coordinates r and 0. we can also describe the ellipse in standard

position with the parametric equations:

Symmetry considerations

� Symmetry considerations can be used to further reduce computations.

� An ellipse in standard position is symmetric between quadrants, but unlike a circle,

it is not symmetric between the two octants of a quadrant.

� Thus, we must calculate pixel positions along the elliptical arc throughout one

quadrant, then we obtain positions in the remaining three quadrants by symmetry

as in the diagram.

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Midpoint Ellipse Algorithm

� we determine points ( x , y) for an ellipse in standard position centered on the

origin.

� And then we shift the points so the ellipse is centered at ( x , y,).

� To display the ellipse in nonstandard position, we could then rotate the ellipse

about its center coordinates to reorient the major and minor axes.

� The midpoint ellipse method is applied throughout the first quadrant in

two parts.

� Following figure shows the division of the first quadrant according to the

slope of an ellipse with rx < ry.

� We define an ellipse function (xc, yc) = (0,0) as

� which has the following properties:

1

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� Thus, the ellipse function fellipse(x, y) serves as the decision parameter in the mid-

point algorithm.

� At each sampling position, we select the next pixel along the ellipse path according

to the sign of the ellipse function evaluated at the midpoint between the two

candidate pixels.

� The ellipse slope is calculated from Eqn 1 as

� At the boundary between region 1 and region 2, dy/dx = - 1 and

� Therefore, we move out of region 1 whenever

� Following figure shows the midpoint between the two candidate pixels at sampling

position xk + 1 in the first region.

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� Assuming position (xk, yk) has been selected at the previous step, we determine

the next position along the ellipse path by evaluating the decision parameter at this

midpoint:

� At the next sampling position (xk+, + 1 = x, + 2), the decision parameter

for region 1 is evaluated as

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� In region 1, the initial value of the decision parameter is obtained by evaluating

the ellipse function at the start position ( x0 , y0) = (0 , r):

� Over region 2, we sample at unit steps in the negative y direction, and the midpoint

is now taken between horizontal pixels at each step.

� For this region, the decision parameter is evaluated as

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� To determine the relationship between successive decision parameters in region 2,

we evaluate the ellipse function at the next sampling step yk+1 - 1=yk - 2

� When we enter region 2, the initial position (x0 , y0) is taken as the last position

selected in region 1 and the initial derision parameter in region 2 is then

ALGORITHM

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

circumference of a ellipse centered on the origin as

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

2. Calculate the initial value of the decision parameters in region 1 as

p0=ry2+rx

2ry+1/4 rx

2.

3. At each xk position in region 1, starting at k=0 perform the following test.

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If pk<0, the next point along the circle centered on (0, 0) is (xk+1, yk) and

P1k+1=p1k+2 ry2xk+1+ ry

2


pk+1=pk+2 ry2xk+1-2 rx

2yk+1+ ry

2

With

2 ry2xk+1=2 ry

2xk+2 and 2 rx

2yk+1=2 rx

2yk+2

And continue until 2 ry2

x>=2 rx2 y

4. Calculate the initial value of the decision parameters in region 2 using the last

point (x0,y0) calculated in region 1 as

P20= ry2

(x0+1/2)2 + rx

2 (y0-1)

2- rx

2 ry

2

5. At each yk position in region 2, starting at k=0, perform the following test.

If p2k>0, the next point along the circle centered on (0, 0) is (xk, yk-1) and

P2k+1=p2k-2 rx2yk+1+ rx

2


P2k+1=p2k +2 ry2xk+1-2 rx

2yk+1+ rx

2

Using the same incremental calculations for x and y as in region 1.

6. Determine symmetry point 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

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A plot of the selected positions around the ellipse boundary within the first

quadrant is shown in Fig. 3-23.

ATTRIBUTES OF OUTPUT PRIMITIVES

� Any parameter that affects the way a primitive is to be displayed is referred to as

an attribute parameter.

� Some attribute parameters, such as

• color and

• size

� Which determine the fundamental characteristics of a primitive.

� Others specify how the primitive is to be displayed under special conditions.

� For example, lines can be

• dotted

• or dashed,

• fat or thin, and

• blue or orange.

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LINE ATTRIBUTES

� Basic attributes of a straight line segment are its

• type,

• its width, and

• its color.

� In some graphics packages, lines can also be displayed using selected pen or brush

options

Line Type

� Possible selections for the line-type attribute include

• solid lines,

• dashed lines,

• and dotted lines

� We modify a line drawing algorithm to generate such lines by setting the length

and spacing of displayed solid sections along the line path.

� To set line type attributes in a program, a user invokes the function

setLineType(lt);

� where parameter lt is assigned a positive integer value of 1,2,3, or 4 to generate

lines that are, solid, dashed, dotted, or dash-dotted respectively.

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Solid Line

Dotted Line

Dashed Line

Dash-Dotted Line

Line Width

� Implementation of line-width options depends on the capabilities of the output

device.

� A heavy line on video monitor could bc displayed as adjacent parallel lines.

� Where as a pen plotter mght require pen changes.

� Line-width command is used to set the current line-width value in the attribute list.

� This value is then used by line-drawing algorithms to Control the thIckness of

lines

� We set the line-wdth attribute with the command:

SetLinewidthScaleFactor(lw);

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� “lw” is assigned a positive number to indicate the relative width of the line to be

displayed.

� A value of 1 specifies a standard-width line.

� Value greater then 1 produce lines thicker than the standard.

� For raster implementation, a standard-width line is generated with single pixels at

each sample position, as in the Bresenham algorithm.

� Other-width lines are displayed by plotting additional pixels along adjacent

parallel line paths.

� Other methods for producing thick Lines include displaying the line as a filled

rectangle or generating the line with a selected pen or brush pattern,

Pen and Brush Options

� With some packages, lines can be displayed with pen or brush selections.

� Options in this category include

• shape,

• size, and

• pattern.

� Some possible pen or brush shapes are given in folwing figure.

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Line Color

� When a system provides color (or intensity) options, a parameter giving the current

color index is included in the list of system-attribute values.

� A polyline routine displays a line in the current color by setting this color value in

the frame buffer at pixel locations along the line path using the setpixel procedure.

� The number of color choices depends on the number of bits available per pixel in

the frame buffer.

� The function is

SetPolylineColorIndex(lc)

� “lc” represents integer values represents the color parameter.

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CURVE ATTRIBUTES

� Parameters for curve attributes are the same as those for line segments.

� We can display curves with varying colors, widths, dot-dash patterns, and

available pen or brush options.

� Methods for adapting curve-drawing algorithms to accommodate attribute

selections are similar to those for line drawing.

� Method for displaying thick curves is to fill in the area between two parallel curve

paths, whose separation distance is equal to the desired width.

COLOR AND GRAYSCALE LEVELS

� Various color and intensity-level options can be made available to a user,

depending of a particular system.

� Options are numerically coded with values ranging from 0 through the positive

integers.

� For CRT monitors, these color codes are then converted to intensitylevel settings

for the electron beams.

� color-information can be stored in the frame buffer in two ways:

• We can store color codes directly in the frame buffer, or

• we can put the color codes in a separate table and use pixel values as an

index into this table

Direct storage scheme

� With the direct storage scheme, whenever a particular color code is specified in an

application program, the corresponding binary value is placed in the frame buffer

for each-component pixel in the output primitives to be displayed in that color.

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� A minimum number of colors can be provided in this scheme with 3 bits of storage

per pixel, as shown in Table.

� Each of the three bit positions is used to control the intensity level (either on or

off) of the corresponding electron gun in an RGB monitor.

• The leftmost bit controls the red gun,

• the middle bit controls the green gun, and

• the rightmost bit controls the blue gun

� Adding more bits per pixel to the frame buffer increases the number of color

choices.

� With 6 bits per pixel, 64 color values are available for each screen pixel.

� With a Resolution of 1024 by 1024, a full-color (24bit per pixel) RGB system

needs 3 mega bytes of storage for the frame buffer.

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� Color tables are an alternate means for providing extended color capabilities,

without requiring large frame buffers.

� In particular, often use color tables to reduce frame-buffer storage requirements.

Grayscale

� With monitors that have no color capability, color functions can be used in an

application program to set the shades of gray, or grayscale, for displayed

primitives.

� Numeric values over the range from 0 to 1 can be used to specify grayscale levels,

which are then converted to appropriate binary codes for storage in the raster.

� This allows the intensity settings to be easily adapted to systems with differing

grayscale capabilities.

� Lists the specifications for intensity codes for a four-level grayscale system.

� In this example, any intensity input value near 0.33 would be stored as the binary

value 01 in the frame buffer, and pixels with this value would be displayed as dark

gray.

� With 3 bits per pixel, we can accommodate 8 gray levels;

� With 8 bits per pixel would give us 256 shades of gray.

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AREA-FILL ATTRIBUTES

� Options for filling a defined region include a choice between a solid color or a

patterned fill.

� These fill options can be applied to polygon regions or to areas defined with

curved boundaries.

� In addition, areas can be painted using various brush styles, colors, and

transparency parameters.

Fill Styles

� Areas are displayed with three basic fill styles:

• hollow with a color border,

• filled with a solid color, or

• filled with a specified pattern or design.

� A basic fill style is selected with the function

� Another value for fill style is hatch, which is used to fill an area with selected

hatching patterns-parallel lines or crossed lines.

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CHARACTER ATTRIBUTES

� The appearance of displayed characters is controlled by attributes such as

• font,

• size,

• color, and orientation.

� Attributes can be set both for entire character strings (text) and for individual

characters defined as marker symbols.

Text Attribute

� There are a great many text options that can be made available to graphics

programmers.

� First of all, there is the choice of font (or typeface).

� which is a set of characters with a particular design style such as

• Arial,

• Courier,

• Impact,

• TimesNewRoman, and various special symbol groups.

� The characters in a selected font can also be displayed with assorted underlining

styles

• Bold face

• Underline

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• Italics

� The corresponding function for setting font is

� SetTextFont();

� Color settings for ,displayed text are stored m the system attribute list.

� SetTextColorIndex(tc)

� Where “tc” specifies the color code.

� We can adjust text size by scaling the overall dimensions (height and width) of

characters or by scaling only the character width.

44 PRABHU.S


2D TRANSFORMATION

� The basic geometric transformations are

• translation,

• rotation, and

• scaling.

� Other transformations that are often applied to objects include

• reflection and

• shear.

Translation

� A translation is applied to an object by repositioning it along a straight-line path

from one coordinate location to another.

� We translate a two-dimensional point by adding translation distances, tx and ty, to

the original coordinate position (x, y) to move the point to a new position ( x ' , y').

� The translation distance pair (tx,ty) is called a translation vector or shift vector.

45 PRABHU.S


� We can express the translation above equations as a single matrix equation by

using

column vectors to represent coordinate positions and the translation vector.

� We can express the translation equations as a single matrix equation by using

column vectors to represent coordinate positions and the translation vector:

� This allows us to write the two-dimensional translation equations in the matrix

form:

� Translation is a rigid-body transformation that moves objects without deformation.

� That is, every point on the object is translated by the same amount.

� A straight Line segment is translated by applying the transformation equation to

each of the line endpoints and redrawing the line between the new endpoint

positions.

� Polygons are translated by adding the translation vector to the coordinate position

of each vertex and regenerating the polygon using the new set of vertex

coordinates and the current attribute settings.

� Following figure illustrates the application of a specified translation vector to

move an object from one position to another.

46 PRABHU.S


� Similar methods are used to translate curved objects.

� To change the position of a circle or ellipse, we translate the center coordinates

and redraw the figure in the new location.

� We translate other curves (for example, splines) by displacing the coordinate

positions defining the objects, then we reconstruct the curve paths using the

translated coordinate points.

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 ө and the position (x1,y1) of the

rotation point (or pivot point) about which the object is to be rotated.

47 PRABHU.S


� Positive values for the rotation angle define counterclockwise rotations about the

pivot point, as in Fig, and negative values rotate objects in the clockwise direction.

� This transformation can also be described as a rotation about a rotation axis that is

perpendicular to the xy plane and passes through the pivot point.

� We first determine the transformation equations for rotation of a point position P

when the pivot point is at the coordinate origin.

� The angular and coordinate relationships of the original and transformed point

positions are shown in Fig.

48 PRABHU.S


� In this figure, r is the constant distance of the point from the origin, angle ф is the

original angular position of the point from the horizontal, and ф is the rotation

angle.

� Using standard trigonometric identities, we can express the transformed

coordinates in terms of angles ө and ф as

� The original coordinates of the point in polar coordinates are

� Substitute equation 2 in 1,

� we can write the rotation equations in the matrix form:

where the rotation matrix is

� When coordinate positions are represented as row vectors instead of column

vectors, the matrix product in rotation equation 4 is transposed so that the

transformed row co ordinate vector [x' y'] is calculated as,

1

2

3

4

49 PRABHU.S


SCALING

A scaling transformation alters the size of an object.

� This operation can be carried out for polygons by multiplying the coordinate

values (x, y) of each vertex by scaling factors sx, and sy, to produce the

transformed coordinates (x', y'):

x' = x. sx , y' = y.sy

� Scaling factor sx, scales objects in the x direction, while sy scales in the y direction.

� The transformation equations 5 can also be written in the matrix form:

Or

� Where S is the 2 by 2 scaling matrix in Eq. 6.

� Any positive numeric values can be assigned to the scaling factors sx, and sy.

� Values less than 1 reduce the size of objects; values greater than 1 produce an

enlargement.

� Specifying a value of 1 for both sx, and sy, leaves the size of objects unchanged.

� When sx, and sy, are assigned the same value, a uniform scaling is produced that

maintains relative object proportions.

� Unique values for sx, and sy, result in a differential scaling.

5

6

7

50 PRABHU.S


� Following figure shows the changing a square (a) into a rectangle (b) with scaling

factors sx, = 2 and sy =1.

� Following figure illustrates scaling a line by assigning the value 0.5 to both sx and

sy, in Eqn 6.

� Both the line length and the distance from the origin are reduced by a factor of 1 /2

MATRIX REPRESENTATIONS AND HOMOGENEOUS COORDINATES

� Many graphics applications involve sequences of geometric transformations.

� An animation, for example, might require an object to be translated and rotated at

each increment of the motion.

51 PRABHU.S


� In design and picture construction applications, we perform

• translations,

• rotations, and

• scaling

to fit the picture components into their proper positions.

� Here we consider how the matrix representations can be used so that such

transformation

sequences can be efficiently processed.

� The basic transformations can be expressed in the general matrix form

� With coordinate positions P and P' represented as column vectors.

� Matrix M1 is a 2 by 2 array containing multiplicative factors, and M2, is a two-

element column matrix containing translational terms.

� For translation, MI is the identity matrix.

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.

� Forming products of transformation matrices is often referred to as a

concatenation, or composition, of matrices.

1

52 PRABHU.S


HOMOGENEOUS COORDINATES

� The term homogeneous is used in mathematics to refer to the effect of this

representation on Cartesian equations.

� when a Cartesian point (x, y) is converted to a homogeneous representation (xh, yh,

h), equations containing x and y, such as f(x, y) = 0.

� Expressing positions in homogeneous coordinates allows us to represent all

geometric transformation equations as matrix multiplications.

� Coordinates are represented with three-element column vectors, and

transformation operations are written as 3 by 3 matrices.

� For translation, we have

� which we can write in the abbreviated form

� with T(tx,ty) as the 3 by 3 translation matrix in the eqn 2.

� Similarly, rotation transformation equations about the coordinate origin are now

written as

Or as

2

3

53 PRABHU.S


� Finally, a scaling transformation relative to the coordinate origin is now expressed

as the matrix multiplication

Or as

OTHER 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.

REFLECTION

� A reflection is a transformation that produces a mirror image of an object.

� The mirror image for a two-dimensional reflection is generated relative to an axis

of reflection by rotating the object 180" about the reflection axis.

54 PRABHU.S


� We can choose an axis of reflection in the xy plane or perpendicular to the xy

plane.

� When the reflection axis is a line in the xy plane, the rotation path about this axis is

in a plane perpendicular to the xy plane.

� For reflection axes that are perpendicular to the xy plane, the rotation path is in the

xy plane.

� Following are examples of some common reflections.

� Reflection about the line y = 0, the x axis, is accomplished with the transformation

matrix

� This transformation keeps x values the same, but "flips" the y values of coordinate

positions.

� The resulting orientation of an object after it has been reflected about the x axis is

shown in Fig.

55 PRABHU.S


� A reflection about the y axis flips x coordinates while keeping y coordinates the

same.

� The matrix for this transformation is

� Following illustrates the change in position of an object that has been reflected

about the line x = 0,

� We flip both the x and y coordinates of a point by reflecting relative to an axis that

is perpendicular to the xy plane and that passes through the coordinate origin.

� This transformation, referred to as a reflection relative to the coordinate origin, has

the matrix representation:

� An example of reflection about the origin is shown in Fig.

56 PRABHU.S


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 a shear.

� Two common shearing transformations are those that shift coordinate x values and

those that shift y values.

� An x-direction shear relative to the x axis is produced with the transformation

matrix

57 PRABHU.S


� which transforms coordinate positions as

� Any real number can be assigned to the shear parameter shx.

� A coordinate position (x, y) is then shifted horizontally by an amount proportional

to its distance (y value) from the x axis (y = 0).

� Setting shx to 2, for example, changes the square in following figure into a

parallelogram.

� Negative values for shx, shift coordinate positions to the left.

� We can generate x-direction shears relative to other reference lines with

� A y-direction shear relative to the line x = x,,+ is generated with the trans-

formation matrix

58 PRABHU.S


TRANSFORMATION FUNCTIONS

� Separate functions are convenient for simple transformation operations, and a

composite function can provide method for specifying complex transformation

sequences.

� Individual commands for generating the basic transformation matrices are

translate (trans-ateVector, matrixTranslate)

rotate (theta, matrixRotate)

scale (scaleVector, matrixScale)

composeMatrix (matrix2, matrix1, matrixOut)

� Each of these functions produces a 3 by 3 transformation matrix that can then be

used to transform coordinate positions expressed as homogeneous column vectors.

� Parameter “translateVector” is a pointer to the pair of translation distances tx and ty.

� Similarly, parameter “scaleVector” specifies the pair of scaling values sx and sy.

� Rotate and scale matrices (matrixTranslate and matrix-Scale) transform with

respect to the coordinate origin.

� A composite transfornation matrix to perform a combination scaling, rotation, and

translation is produced with the function

buildTransformationMatrix (referencepoint, translatevector, theta, scalevector,

matrix)

59 PRABHU.S


TWO DIMENSIONAL VIEWING

� A graphics package allows a user to specify which part of a defined picture is to be

display& and where that part is to be placed on the display device.

� Transformations from world to device coordinates involve translation, rotation,

and scaling operations, as well as procedures for deleting those parts of the picture

that are outside the limits of a selected display area.

� A world-coordinate area selected for display is called a window.

� An area on a display device to which a window is mapped is called a viewport.

� The window defines what is to be viewed; the viewport defines where it is to be

displayed.

� In general, the mapping of a part of a world-coordinate scene to device coordinates

is referred to as a viewing transformation.

� Sometimes the two-dimensional viewing transformation is simply referred to as

the window-to-viewport transformation or the windowing transformation.

� Following figure illustrates the mapping of a picture section that falls within a

rectangular window onto a designated rectangular viewport.

60 PRABHU.S


Viewing-Transformation

� Some graphics packages that provide window and viewport operations allow only

standard rectangles.

� But a more general approach is to allow the rectangular window to haw any

orientation.

� In this case, we carry out the viewing transformation in several steps, as indicated

in Fig.

� First, we construct the scene in world coordinates using the output primitives and

attributes.

61 PRABHU.S


� Next. to obtain a particular orientation for the window, we can set up a two-

dimensional viewing-coordinate system in the world-coordinate plane, and define

a window in the viewing-coordinate system.

� The viewing coordinate reference frame is used to provide a method for setting up

arbitrary

� Orientations for rectangular windows.

� Once the viewing reference frame is established, we can transform descriptions in

world coordinates to viewing coordinates.

� We then define a viewport in normalized coordinates (in the range from 0 to 1 )

and map the viewing-coordinate description of the scene to normalized

coordinates.

� At the final step, all parts of the picture that he outside the viewport are clipped,

and the contents of the viewport are transferred to device coordinates.

� Following figure i1lustratt.s a rotated viewing-coordinate reference frame and the

mapping to normalized coordinates.

62 PRABHU.S


WINDOW-TO-VIEWPORT COORDINATE TRANSFORMATION

� Once object descriptions have been transferred to the viewing reference frame, we

choose the window extents in viewing coordinates and select the viewport limits in

normalized coordinates.

� Object descriptions are then transferred to normalized device coordinates.

� We do this using a transformation that maintains the same relative placement of

objects in normalized space as they had in viewing coordinates.

� If a coordinate position is at the center of the viewing window, for instance, it will

be displayed at the center of the viewport.

� Following figure illustrates the window-to-viewport mapping.

� A point at position (xw, yw) in the window is mapped into position (xv, yv) in the

associated viewport.

� To maintain the same relative placement in the viewport as in the window, we

require that

63 PRABHU.S


� Solving these expressions for the viewport position (xv, yv), we have

� where the scaling factors are

� Above equations can also be derived with a set of transformations that converts the

window area into the viewport area.

� This conversion is performed with the following sequence of transformations:

1. Perform a scaling transformation using a fixed-point position of (xw,yw) that

scales the window area to the size of the viewport.

2. Translate the scaled window area to the position of the viewport.

64 PRABHU.S


CLIPPING OPERATIONS

� Generally, any procedure that identifies those portions of a picture that are either

inside or outside of a specified region of space is referred to as a clipping

algorithm, or simply clipping.

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

� For the viewing transformation, we want to display only those picture parts that are

within the window area.

� Everything outside the window is discarded.

� Clipping algorithms can be applied in world coordinates, so that only the contents

of the window interior are mapped to device coordinates.

� Alternatively, the complete world-coordinate picture can be mapped first to device

coordinates, or normalized device coordinates, then clipped against the viewport

boundaries.

� we consider algorithms for clipping the following primitive types

• Point Clipping

• Line Clipping (straight-line segments)

• Area Clipping (polygons)

• Curve Clipping

• Text Clipping

65 PRABHU.S


LINE CLIPPING

� Following figure illustrates possible relationships between line positions and a

standard

rectangular clipping region.

� A line clipping procedure involves several parts.

� First, we can test a given line segment to determine whether it lies completely

inside the clipping window.

� If it does not, we try to determine whether it lies completely outside the window.

� Finally, if we cannot identify a line as completely inside or completely outside, we

must perform intersection calculations with one or more clipping boundaries.

� We process lines through the "inside-outside'' tests by checking the line endpoints.

� A line with both endpoints inside all clipping boundaries, such as the line from P1,

to P2, is saved.

� A line with both endpoints outside any one of the clip boundaries (line P3P4 in

above Fig.) is outside the window.

66 PRABHU.S


� All other lines cross one or more clipping boundaries, and may require calculation of

multiple intersection points.

� To minimize calculations, we try to devise clipping algorithms that can efficiently

identify outside lines and reduce intersection calculations.

� For a line segment with endpoints (x1, y1) and (x2, y2) and one or both endpoints

outside the clipping rectangle.

x = x1 + u(x2 - x1)

y = y1 + u(y2 - y1) 0≤u≤1

� The parametric representation could be used to determine values of parameter u for

intersections with the clipping boundary coordinates.

� If the value of u for an intersection with a rectangle boundary edge is outside the

range 0 to 1, the line does not enter the interior of the window at that boundary.

� If the value of u is within the range from 0 to 1, the line segment does indeed cross

into the clipping area.

� This method can be applied to each clipping boundary edge in turn to determine

whether any part of the line segment is to be displayed.

� Line segments that are parallel to window edges can be handled as specia1 cases.

� Clipping line segments with these parametric tests requires a good deal of computation,

and faster approaches to clipping are possible.

� A number of efficient line clippers have been developed.

67 PRABHU.S


COHEN-SUTHERLAND LINE CLIPPING

� This is one of the oldest and most popular line-clipping procedures.

� Generally, the method speeds up the processing of line segments by performing

initial tests

that reduce the number of intersections that must he calculated.

� Every line end point in a picture is assigned a four-digit binary code, called a

region code,

� That identifies the location of the point relative to the boundaries of the clipping

rectangle.

� Regions are set up in reference to the boundaries as shown in Following fig.

� Each bit position in the region code is used to indicate one of the four relative

coordinate

positions of the point with respect to the clip window:

• to the left,

• right,

• top, or

• bottom.

� By numbering the bit positions in the region code as 1 through 4 from right to left,

the co ordinate regions can be correlated with the bit positions as

68 PRABHU.S


bit 1: left

bit 2: right

bit 3: below

bit 4: above

� A value of 1 in any bit position indicates that the point is in that relative position;

� otherwise, the bit position is set to 0.

� If a point is within the clipping rectangle, the region code is 0000.

� A point that is below and to the left of the rectangle has a region code of 0101.

� Bit values in the region code are determined by comparing endpoint coordinate

values (x, y) to the clip boundaries.

� Bit 1 is set to 1 if xwmin .

� The other three bit values can be determined using similar comparisons.

� For languages in which bit manipulation is possible, region-code bit values can be

determined

with the following two steps:

(1) Calculate differences between endpoint coordinates and clipping boundaries.

(2) Use the resultant sign bit of each difference calculation to set the corresponding

value in

the region code.

• Bit 1 is the sign bit of x-xwmin;

• Bit 2 is the sign bit of xwmax-x;

• bit 3 is the sign bit of y-ywmin;

• bit 4 is the sign bit of ywmax-y;

� Once we have established region codes for all line endpoints, we can quickly

determine which lines are completely inside the clip window and which are clearly

outside.

69 PRABHU.S


� Any lines that are completely contained within the window boundaries have a

region code of 0000 for both endpoints, and we accept these lines.

� Any lines that have a 1 in the same bit position in the region codes for each

endpoint are completely outside the clipping rectangle, and we reject these lines.

� We would discard the line that has a region code of 1001 for one endpoint and a

code of 0101 for the other endpoint.

� Both endpoints of this line are left of the clipping rectangle, as indicated by the 1

in the first bit position of each region code.

� A method that can be used to test lines for total clipping is to perform the logical

and operation with both region codes.

� If the result is not 0000, the line is completely outside the clipping region.

� Lines that cannot be identified as completely inside or completely outside a clip

window by these tests are checked for intersection with the window boundaries.

� As shown in figure, such lines may or may not cross into the window interior.

70 PRABHU.S


� We begin the clipping process for a line by comparing an outside endpoint to a

clipping boundary to determine how much of the line can be discarded.

� Then the remaining part of the Line is checked against the other boundaries, and

we continue until either the line is totally discarded or a section is found inside the

window.

� We set up our algorithm to check line endpoints against clipping boundaries in the

order left, right, bottom, top.

POLYGON CLIPPING

� To clip polygons, we need to modify the line-clipping procedures.

� 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.

� What we really want to display is a bounded area after clipping, as in Fig.

71 PRABHU.S


� For polygon clipping, we require an algorithm that will generate one or more

closed areas that are then scan converted for the appropriate area fill.

� The output of a polygon clipper should be a sequence of vertices that defines the

clipped polygon boundaries.

Sutherland-Hodgeman Polygon Clipping

� We can correctly clip a polygon by processing the polygon boundary as a whole

against each window edge.

� This could be accomplished by processing all polygon vertices against each clip

rectangle boundary in turn.

� Beginning with the initial set of polygon vertices, we could first clip the polygon

against the left rectangle boundary to produce a new sequence of vertices.

� The new set of vertices could then k successively passed to a right boundary

clipper, a bottom boundary clipper, and a top boundary clipper, as in Fig.

72 PRABHU.S


� At each step, a new sequence of output vertices is generated and passed to the next

window boundary clipper.

� There are four possible cases when processing vertices in sequence around the

perimeter of a polygon.

� As each pair of adjacent polygon vertices is passed to a window boundary clipper,

we make the following tests:

1. If the first vertex is outside the window boundary and the second vertex is

inside, both the intersection point of the polygon edge with the window

boundary and the second vertex are added to the output vertex list.

2. If both input vertices are inside the window boundary, only the second vertex is

added to the output vertex list.

3. If the first vertex is inside the window boundary and the second vertex is

outside, only the edge intersection with the window boundary is added to the

output vertex list.

4. If both input vertices are outside the window boundary, nothing is added to the

output list.

� These four cases are illustrated in following figure for successive pairs of polygon

vertices.

73 PRABHU.S


� Once all vertices have been processed for one clip window boundary, the output

11st of vertices is clipped against the next window boundary.

� We illustrate this method by processing the area in following figure against the left

window boundary.

� Vertices 1 and 2 are found to be on the outside of the boundary.

� Moving along to vertex 3, which is inside, we calculate intersection and save both

the intersection point and vertex 3.

� Vertices 4 and 5 are determined to be inside, and they also are saved.

� Moving along to vertex 6 from 5, we need to find the intersection and it is saved.

� Using the five saved points, we would repeat the process for the next window

boundary.

74 PRABHU.S


CURVE CLIPPING

� Areas with curved boundaries can be clipped with methods similar to those

discussed in the line clipping.

� Curve-clipping procedures will involve nonlinear equations, however, and this

requires more processing than for objects with linear boundaries.

� The bounding rectangle for a circle or other curved object can be used first to test

for overlap with a rectangular clip window.

� If the bounding rectangle for the object is completely inside the window, we save

the object.

� If the rectangle is determined to be completely outside the window, we discard the

object.

� In either case, there is no further computation necessary.

� But if the bounding rectangle test fails, we can look for other computation saving

approaches.

� For a circle, we can use the coordinate extents of individual quadrants and then

octants for preliminary testing before calculating curve-window intersections.

� For an ellipse, we can test the coordinate extents of individual quadrants.

� Following figure illustrates circle clipping against a rectangular window.

75 PRABHU.S


� Similar procedures can be applied when clipping a curved object against a general

polygon clip region.

� On the first pass, we can clip the bounding rectangle of the object against the

bounding rectangle of the clip region.

� If the two regions overlap, we will need to solve the simultaneous line-curve

equations to obtain the clipping intersection points.

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.

The simplest method for processing character strings relative to a window boundary is

to use the all-or-none string-clipping strategy shown in Fig.

� If all of the string is inside a clip window, we keep it.

� Otherwise, the string is discarded.

� This procedure is implemented by considering a bounding rectangle around the

text pattern.

76 PRABHU.S


� The boundary positions of the rectangle are then compared to the window

boundaries, and the string is rejected if there is any overlap.

� This method produces the fastest text clipping.

� An alternative to rejecting an entire character string that overlaps a window

boundary is to use the all-or-none character-clipping strategy.

� Here we discard only those characters that are not completely inside the window.

In this case, the boundary limits of individual characters are compared to the window.

� Any character that either overlaps or is outside a window boundary is clipped.

� A final method for handling text clipping is to clip the components of individual

characters.

� We now treat characters in much the same way that we treated lines.

� If an individual character overlaps a clip window boundary, we clip off the parts of

the character that are outside the window.

77 PRABHU.S


� Outline character fonts formed with line segments can be processed in this way

using a line clipping algorithm.

� Characters defined with bit maps would be clipped by comparing the relative

position of the individual pixels in the character grid patterns to the clipping

boundaries.

EXTERIOR CLIPPING

� we have considered only procedures for clipping a picture to the interior of a region

by eliminating everything outside the clipping region.

� What is saved by these procedures is inside the region.

� In some cases, we want to do the reverse, that is, we want to clip a picture to the

exterior of a specified region.

� The picture parts to be saved are those that are outside the region.

� This is referred to as exterior clipping.

� A typical example of the application of exterior clipping is in multiple window

systems.

� To correctly display the screen windows, we often need to apply both internal and

external clipping.

� Following figure illustrates a multiple window display.

78 PRABHU.S


� Objects within a window are clipped to the interior of that window.

� When other higher-priority windows overlap these objects, the objects are also

clipped to the exterior of the overlapping windows.

� Exterior clipping is used also in other applications that require overlapping

pictures.

� Examples here include the design of page layouts in advertising or publishing

applications or for adding labels or design patterns to a picture.

� The technique can also be used for combining graphs, maps, or schematics.

� For these applications, we can use exterior clipping to provide a space for an insert

into a larger picture.

79 PRABHU.S


UNIT II

3D CONCEPTS

� To obtain a display of a three-dimensional scene that has been modeled in world

coordinates.

� we must first set up a coordinate reference for the "camera".

� This coordinate reference defines the position and orientation for the plane of the

camera film.

Which is the plane we want to use to display a view of the objects in the scene?

� Object descriptions are then transferred to the camera reference coordinates and

projected onto the selected display plane.

� We can then display the objects in wireframe (outline) form, as in Fig,

PRABHU.S


� Or we can apply lighting and surface render

surfaces.

Parallel Projection

� One method for generating a view of a solid object is to project points on the

object surface along parallel

� By selecting different viewing

object onto the display plane to obtain different two

object, as in Fig.

� In a parallel projection, parallel lines in the world

parallel

lines on the two-dimensional


apply lighting and surface rendering techniques to shade the visible

One method for generating a view of a solid object is to project points on the

surface along parallel lines onto the display plane.

By selecting different viewing positions, we can project visible points on the

object onto the display plane to obtain different two-dimensional views of the

parallel lines in the world-coordinate scene project into

dimensional display plane.

80

ing techniques to shade the visible

One method for generating a view of a solid object is to project points on the

project visible points on the

dimensional views of the

coordinate scene project into

81 PRABHU.S


� This technique is used in engineering and architectural drawings to represent an

object with a set of views that maintain relative proportions of the object.

� The appearance of the solid object can then be reconstructured from the major

views.

Perspective Projection

Perspective : “The appearance of things relative to one another as determined by their

distance from the viewer”

� Another method for generating a view of a three-dimensional scene is to project points

to the display plane along converging paths.

� This causes objects farther from the viewing position to be displayed smaller than

objects of the same size that are nearer to the viewing position.

� In a perspective projection, parallel lines in a scene that are not parallel to the display

plane are projected into converging lines.

� Scenes displayed using perspective projections appear more realistic, since this is the

way that our eyes and a camera lens form images.

� In the perspective projection view shown in Fig.

82 PRABHU.S


� Parallel lines appear to converge to a distant point in the background, and distant objects

appear smaller than objects closer to the viewing position.

Depth Cueing

� Depth information is important so that we can easily identify, for a particular

viewing direction, which is the front and which is the back of displayed objects.

� Following figure illustrates the ambiguity that can result when a wireframe object

is displayed without depth information.

� The wireframe representation of the pyramid in

(a) Contains no depth information to indicate whether the viewing direction is

(b) Downward from a position above the apex or

(c) Upward from a position below the base.

� There are several ways in which we can include depth information in the two-

dimensional representation of solid objects.

� A simple method for indicating depth with wireframe displays is to vary the intensity of

objects according to their distance from the viewing position.

� Following figure shows a wireframe object displayed with depth cueing.

83 PRABHU.S


� The lines closest to the viewing position are displayed with the highest intensities,

and lines farther away are displayed with decreasing intensities.

PROJECTIONS

� Once world-coordinate descriptions of the objects in a scene are converted to

viewing coordinates, we can project the three-dimensional objects onto the two

dimensional view plane.

� There are two basic projection methods.

• Parallel Projection

• Perspective Projection

Parallel Projection

� In a parallel projection, coordinate positions are transformed to the view plane

along parallel Line.

84 PRABHU.S


� A parallel projection preserves relative proportions of objects.

� This is the method used in drafting to produce scale drawings of three-dimensional

objects.

� Accurate views of the various sides of an object are obtained with a parallel projection.

� But this does not give us a realistic representation of the appearance of a 3D

dimensional object.

Orthographic parallel projection.

� We can specify a parallel projection with a projection vector that defines the

direction for the projection lines.

� When the projection is perpendicular to the view plane, we have an orthographic

parallel projection.

� Otherwise, we have an oblique parallel projection.

� Following figure illustrates the two types of parallel projections.

85 PRABHU.S


� Some graphics packages, such as GL on Silicon Graphics workstations, do not

provide for oblique projections.

� In this package, for example, a parallel projection is specified by simply giving the

boundary edges of a rectangular parallelepiped.

� Orthographic projections are most often used to produce the front, side, and

top view of an object, a s shown in Fig.

86 PRABHU.S


� Front, side, and rear orthographic projections of an object are called elevations.

� And a top orthographic projection is called plan view.

� Engineering and architectural drawings commonly employ these orthographic

projections, because lengths and angles are accurately depicted and can be

measured from the drawings.


� For a perspective projection, object positions are transformed to the view plane

along lines that converge to a point called the projection reference point (or center

of projection).

� A perspective projection, on the other hand, 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

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� The projected view of an object is determined by calculating the intersection of the

projection lines with the view plane.

� To obtain a perspective projection of a three-dimensional object, we transform

points along projection lines that meet at the projection reference point.

� Suppose we set the projection reference point at position zprp along the zv axis, and

we place the view plane at as shown in Fig.

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� We can write equations describing coordinate positions along this perspective

projection line in parametric form as

� Parameter u takes values from 0 to 1.

� Coordinate position (x', y', z') represents any point along the projection line.

� When u = 0, we are at position P = (x , y, z).

� At the other end of the line, u = 1 and we have the projection reference point

coordinates (0, 0, zprp).

� On the view plane, z' = zprp and we can solve the z' equation for parameter u at this

position along the projection line:

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3D REPRESENTATION

� Representation schemes for solid objects are often divided into two broad

categories,

1. Boundary representations

2. Space-partitioning representation

Boundary representations

Boundary representations (B-reps) describe a three-dimensional object as a set of

surfaces that separate the object interior from the environment.

Typical examples of boundary representations are polygon facets and spline patches.

Space-partitioning representation

Space-partitioning representations are used to describe interior properties, by

partitioning the spatial region containing an object into a set of small, non

overlapping, contiguous solids (usually cubes).

A common space-partitioning description for a three-dimensional object is an octree

representation.

POLYGON SURFACES

� The most commonly used boundary representation for a three-dimensional

graphics object is a set of surface polygons that enclose the object interior.

� Many graphics systems store all object descriptions as sets of surface polygons.

� This simplifies and speeds up the surface rendering and display of objects, since all

surfaces are described with linear equations.

� For this reason, polygon descriptions are often referred to as "standard graphics

objects."

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� In some cases, a polygonal representation is the only one available, but many

packages allow objects to be described with other schemes, such as spline surfaces,

that are then converted to polygonal representations for processing.

� A polygon representation for a polyhedron precisely defines the surface features of

the object.

� But for other objects, surfaces are tesselated (or tiled) to produce the polygon-

mesh approximation.

� Following figure shows Wireframe representation of a cylinder with back (hidden

lines removed).

� Such representations are common in design and solid-modeling applications,

since the wireframe outline can be displayed quickly to give a general

indication of the surface structure.

� Realistic renderings are produced by interpolating shading patterns across the

polygon surfaces to eliminate or reduce the presence of polygon edge

boundaries.

� And the polygon-mesh approximation to a curved surface can be improved by

dividing the surface into smaller polygon facets.

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Polygon Tables

� We specify a polygon surface with a set of vertex coordinates and associated

attribute

parameters.

� An information for each polygon are placed into tables that are to be used in the

subsequent processing, display, and manipulation of the objects in a scene.

� Polygon data tables can be organized into two groups:

1. geometric tables and

2. attribute tables.

Geometric tables

� It contain vertex coordinates and parameters to identify the spatial orientation of

the polygon surfaces.

Attribute tables

� It includes parameters specifying the degree of transparency of the object and its

surface reflectivity and texture characteristics.

� A convenient organization for storing geometric data is to create three lists:

1. a vertex table,

2. an edge table, and

3. a polygon table.

Vertex table

� Coordinate values for each vertex in the object are stored in the vertex table.

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Edge table

� The edge table contains pointers back into the vertex table to identify the vertices

for each polygon edge.

Polygon table

� The polygon table contains pointers back into the edge table to identify the edges

for

each polygon.

� This scheme is illustrated in Fig for two adjacent polygons on an object surface.

tesselated : Fit together exactly, of identical shapes

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� In addition, individual objects and their component polygon faces can be assigned

object and facet identifiers for easy reference.

Plane Equations

� To produce a display of a three-dimensional object, we must process the input data

representation for the object through several procedures.

� These processing steps include transformation of the modeling and world-coordinate

descriptions to viewing coordinates, then to device coordinates; identification of visible

surfaces; and the application of surface-rendering procedures.

� For some of these processes, we need information about the spatial orientation of

the individual surface components or the object.

� This information is obtained from the vertex coordinate values and the equations

that describe the polygon planes.

� The equation for a plane surface can be expressed in the form

� where (x, y, z ) in any point on the plane, and the coefficients A, B, C, and D are

� constants describing the spatial properties of the plane.

� We can obtain the values of A, B, C, and D by solving a set of three plane

equations.

� To solve the following set of simultaneous linear plane equations for the ratios

A/D, B/D,and C/D:

AX + BY + CZ + D = 0

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� The solution for this set of equations can be obtained in determinant form, using

Cramer's rule, as

� Expanding the determinants, we can write the calculations for the plane

coefficients in the form

POLYGON MESHES

� Some graphics packages provide several polygon functions for modeling objects.

� A single plane surface can be specified with a function such as “fillArea”.

� But when object surfaces are to be tiled, it is more convenient to specify the

surface facets with a mesh function.

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Triangle strip

� One type of polygon mesh is the triangle strip.

� This function produces n - 2 connected triangles, .as shown in above figure.

Quadrilateral mesh

� Another similar function is the quadrilateral mesh.

� which generates a mesh of (n - 1) by (m - 1) quadrilaterals, given the coordinates

for an n by m array of vertices.

� Above figure shows 20 vertices forming a mesh of 12 quadrilaterals.

Problem

� When polygons are specified with more than three vertices, it is possible that the

vertices may not all Lie in one plane.

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� This can be due to numerical errors or errors in selecting coordinate positions for

the vertices.

Solution

� One way to handle this situation is simply to divide the polygons into triangles.

� Another approach that is sometimes taken is to approximate the plane parameters

A, B, and C.

� We can do this with averaging methods or we can project the polygon onto the

coordinate planes.

� Using the projection method, we take

A proportional to the area of the polygon projection on the yz plane,

B proportional to the projection area on the xz plane, and

C proportional to the projection area on the xy plane.

CURVED LINES AND SURFACES

� Displays of three dimensional curved lines and surfaces can be generated from an

input set of mathematical functions defining the objects or from a set of user

specified data points.

� When functions are specified, a package can project the defining equations for a

curve to the display plane and plot pixel positions along the path of the projected

function.

� For surfaces, a functional description is often tesselated to produce a polygon-

mesh approximation to the surface.

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� Usually, this is done with triangular polygon patches to ensure that all vertices of

any polygon are in one plane.

� Polygons specified with four or more vertices may not have all vertices in a single

plane.

� Curve and surface equations can be expressed in either a parametric or a non

parametric form.

QUADRIC SUKFACES

� A frequently used class of objects are the quadric surfaces, which are described

with second-degree equations (quadratics).

� They include

• spheres,

• ellipsoids,

• tori,

• paraboloids, and

• hyperboloids.

� Quadric surfaces, particularly spheres and ellipsoids, are common elements of

graphics scenes, and they are often available in graphics packages as primitives

from which more complex objects can be constructed.

SPHERE

� In Cartesian coordinates, a spherical surface with radius r centered on the

coordinate origin is defined as the set of points (x, y, z) that satisfy the equation

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� We can also describe the spherical surface in parametric form, using latitude and

longitude angles.

� The above figure shows the Parametric coordinate position (r, θ,ф) on the surface

of a sphere with radius r.

ELLIPSOID

� An ellipsoidal surface can be described as an extension of a spherical surface,

where the radii in three mutually perpendicular directions can have different

values.

2

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� The Cartesian representation for points over the surface of an ellipsoid centered on

the origin is

� And a parametric representation for the ellipsoid in terms of the latitude angle θ

and the longitude angle ф in eqn 2.

SPLINE

� A spline is a flexible strip used to produce a smooth curve through a designated set

of points.

� Several small weights are distributed along the length of the strip to hold it in

position on the drafting table as the curve is drawn.

� The term spline curve originally referred to a curve drawn in this manner.

� In computer graphics, the term spline curve refers to any composite curve formed

with polynomial sections satisfying specified continuity conditions at the boundary

of the pieces.

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� Splines are used in graphics applications to design curve and surface shapes, to

digitize drawings for computer storage, and to specify the animation paths for the

objects or the camera in a scene.

� Typical CAD applications for splines include the design of automobile bodies,

aircraft and spacecraft surfaces, and ship hulls.

� The above figure shows the set of six control points interpolated with piecewise

continuous polynomial.

Spline Specifications

� There are three equivalent methods for specifying a particular spline

representation:

1. We can state the set of boundary conditions that are imposed on the spline; or

2. We can state the matrix that characterizes the spline; or

3. We can state the set of blending functions (or basis functions) that determine

how specified geometric constraints on the curve are combined to calculate

positions along the curve path.

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VISUALIZATION OF DATA SETS

� The use of graphical methods as an aid in scientific and engineering analysis is

commonly referred to as scientific visualization.

� This involves the visualization of data sets and processes that may be difficult or

impossible to analyze without graphical methods.

� For example, visualization techniques are needed to deal with the output of high-

volume data sources such as

• supercomputers,

• satellite, and spacecraft scanners,

• radio-astronomy telescopes, and

• medical scanners.

� Similar methods employed by commerce, industry, and other nonscientific areas

are sometimes referred to as business visualization.

� Data sets are classified according to their spatial distribution and according to data

type.

� Two-dimensional data sets have values distributed over a surface, and three-

dimensional data sets have values distributed over the interior of

• a cube,

• a sphere, or

• some other region of space.

� Data types include

• scalars,

• vectors,

• tensors, and

• multivariate data.

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Visual Representations for Scalar Fields

� A scalar quantity is one that has a single value.

� Scalar data sets contain values that may be distributed in time, as well as over

spatial positions.

� Also, the data values may be functions of other scalar parameters.

� Some examples of physical scalar quantities are

• energy,

• density,

• mass,

• temperature,

• pressure,

• charge,

• resistance,

• reflectivity, and

• frequency.

� A common method for visualizing a scalar data set is to use graphs or charts that

show the distribution of data values.

� If the data are distributed over a surface, we could plot the data values as vertical

bars rising up from the surface, or we can interpolate the data values to display a

smooth surface.

Pseudo-color methods

� Pseudo-color methods are also used to distinguish different values in a scalar data

set, and color-coding techniques can be combined with graph and chart methods.

� To color code a scalar data set, we choose a range of color and map the range of

data values to the color range.

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� For example, blue could be assigned to the lowest scalar value, and red could be

assigned to the highest value.

� Following figure gives an example of a color-coded surface plot.

� Color coding a data set can be tricky, because some color combinations can lead to

misinterpretations of the data.

� Contour plots are used to display isolines (lines of constant scalar value) for a

data set distributed over a surface.

� The isolines are spaced at some convenient interval to show the range and

variation of the data values over the region of space.

� The isolines are usually plotted as straight-line sections across each cell, as

illustrated in Fig.

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Visual Representations for Vector Fields

� A vector quantity V in three-dimensional space has three scalar values ( Vx , Vy,

Vz) one for each coordinate direction, and a two-dimensional vector has two

components (Vx, Vy).

� Another way to describe a vector quantity is by giving its magnitude |V| and its

direction as a unit vector u.

� As with scalars, vector quantities may be functions of position, time, and other

parameters.

� Some examples of physical vector quantities are

• velocity,

• acceleration,

• force,

• electric fields,

• magnetic fields,

• gravitational fields, and

• electric current.

� One way to visualize a vector field is to plot each data point as a small arrow that

shows the magnitude and direction of the vector.

� This method is most often used with cross-sectional slices, as in Fig.

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� Magnitudes for the vector values can be shown by varying the lengths of the

arrows, or we can make all arrows the same size, but make the arrows different

colors according to a selected color coding for the vector magnitudes.

� We can also represent vector values by plotting field lines or streamlines.

� Field lines are commonly used for electric, magnetic, and gravitational fields.

� The magnitude of the vector values is indicated by the spacing between field lines, and

the direction is the tangent to the field, as shown in Fig

� Streamlines can be displayed as wide arrows.

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Visual Representations for Tensor Fields

� A tensor quantity in three-dimensional space has nine components and can be

represented with a 3 by 3 matrix.

� Actually, this representation is used for a second-order tensor, and higher-order

tensors do occur in some applications, particularly general relativity.

� Some examples of physical, second-order tensors are stress and strain in a material

subjected to external forces, conductivity (or resistivity) of an electrical conductor,

and the metric tensor, which gives the properties of a particular coordinate space.

� The stress tensor in Cartesian coordinates, for example, can be represented as

� Tensor quantities are frequently encountered in anisotropic materials, which have

different properties in different directions.

Visual Representations for Multivariate Data Fields

� In some applications, at each grid position over some region of space, we may

have multiple data values.

� Which can be a mixture of scalar, vector, and even tensor values.

� As an example, for a fluid-flow problem, we may have fluid velocity, temperature,

and density values at each three-dimensional position.

� Thus, we have five scalar values to display at each position, and the situation is

similar to displaying a tensor field.

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� A method for displaying multivariate data fields is to construct graphical objects,

sometimes referred to as glyphs, with multiple parts.

� Each part of a glyph represents a physical quantity.

� The size and color of each part can be used to display information about scalar

magnitudes.

� To give directional information for a vector field, we can use a wedge, a cone, or

some other pointing shape for the glyph part representing the vector.

� An example of the visualization of a multivariate data field using a glyph structure

at selected grid positions is shown in

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3D TRANSFORMATION

� Methods for geometric transformations and object modeling in three dimensions

are extended from two-dimensional methods by including considerations for the z

coordinate.

TRANSLATION

� In a three-dimensional homogeneous coordinate representation, a point is

translated

from position P = (x, y, z) to position P' = (x', y', z') with the matrix operation

Or

1

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� Parameters tx, ty, and tz, specifying translation distances for the coordinate directions x,

y, and z, are assigned any real values.

� The matrix representation in Eq.1 is equivalent to the three equations

� An object is translated in three dimensions by transforming each of the defining

points of the object.

� For an object represented as a set of polygon surfaces, we translate each vertex of

each surface and redraw the polygon facets in the new position.

� We obtain the inverse of the translation matrix in Eq.1 by negating the translation

distances tx, ty, and tz.

� This produces a translation in the opposite direction, and the product of a translation

matrix and its inverse produces the identity matrix.

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.

� Following figures illustrate Positive rotation directions about the coordinate axes

are

counterclockwise, when looking toward the origin from a positive coordinate

position on each axis.

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Coordinate-Axes Rotations

� The two-dimensional z-axis rotation equations are easily extended to three

dimensions:

� Parameter θ specifies the rotation angle.

� In homogeneous coordinate form, the three-dimensional z-axis rotation equations are

expressed as

� which we can write more compactly as

� Following figure illustrates rotation of an object about the z axis.

1

2

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� Transformation equations for rotations about the other two coordinate axes can be obtained

with a cyclic permutation of the coordinate parameters x, y and in Eqs.1.

� That is, we use the replacements

� as illustrated in following fig.

� Substituting permutations 3 in Eqs. 1, we get the equations for an x-axis rotation:

� Which can be written in the homogeneous coordinate form

3

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� Cyclic permutation of the Cartesian-coordinate axes to produce the three sets of

coordinate axis rotation equations.

SCALING

� The matrix expression tor the scaling transformation of a position P = (x, y, z)

relative

to the coordinate origin can be written as

Or

� Where scaling parameters sx, sy, and sz are assigned any positive values.

� Explicit expressions for the coordinate transformations for scaling relative to the

origin are

� Scaling an object with transformation Eqn1 changes the size of the object and

repositions the object relative to the coordinate origin.

� Also, if the transformation parameters are not all equal, relative dimensions in the

object are changed.

1

2

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� We preserve the original shape of an object with a uniform scaling (sx =sy = sz).

� The result of scaling an object uniformly with each scaling parameter set to 2 is

shown in Fig.

� Scaling with respect to a selected fixed position (xf, yf, zf) can be represented with

the following transformation sequence:

1. Translate the fixed point to the origin.

2. Scale the object relative to the coordinate origin using Eq1.

3. Translate the fixed point back to its original position.

� This sequence of transformations is demonstrated in following fig.

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� The matrix representation for an arbitrary fixed-point scaling can then be

expressed as the concatenation of these translate-scale-translate transformations as

� We form the inverse scaling matrix for either Eqn1 or Eqn3 by replacing the

scaling parameters

sx, sy and sz with their reciprocals.

� The inverse matrix generates an opposite scaling transformation, so the

concatenation of any

scaling matrix and its inverse produces the identity matrix.

OTHER TRANSFORMATIONS

� In addition to translation, rotation, and scaling, there are various additional

transformations

that are often useful in three-dimensional graphics applications.

� Two of these are

• reflection and

• shear.

3

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REFLECTIONS

� A three-dimensional reflection can be performed relative to a selected reflection

axis or with respect to a selected reflection plane.

� In general, three-dimensional reflection matrices are set up similarly to those for

two dimensions.

� Reflections relative to a given axis are equivalent to 1800 rotations about that axis.

� Reflections with respect to a plane are equivalent to 180' rotations in four-

dimensional space.

� When the reflection plane is a coordinate plane (either xy, xz, or yz), we can think

of the transformation as a conversion between Left-handed and right-handed

systems.

� An example of a reflection that converts coordinate specifications from a right-

handed system

to a left-handed system (or vice versa) is shown in Fig.

� This transformation changes the sign of the z coordinates, leaving the x and y-

coordinate

values unchanged.

� The matrix representation for this reflection of points relative to the xy plane is

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� Transformation matrices for inverting x and y values are defined similarly, as

reflections relative

to the yz plane and xz plane, respectively.

� Reflections about other planes can be obtained as a combination of rotations and

coordinate-plane reflections.

SHEARS

� Shearing transformations can he used to modify object shapes.

� They are also useful in three-dimensional viewing for obtaining general projection

transformations.

� In two dimensions, we discussed transformations relative to the x or y axes to

produce distortions in the shapes of objects.

� In three dimensions, we can also generate shears relative to the z axis.

� As an example of three-dimensional shearing. the following transformation

produces a z-axis shear:

� Parameters a and b can be assigned any real values.

� The effect of this transformation matrix is to alter x- and y-coordinate values by an

amount

that is proportional to the z value, while leaving the z coordinate unchanged.

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� Boundaries of planes that are perpendicular to the z axis are thus shifted by an

amount proportional to z.

� An example of the effect of this shearing matrix on a unit cube is shown in Fig, for

shearing values a = b =1.

� Shearing matrices for the x axis and y axis are defined similarly.

VIEWING PIPELINE

� The steps for computer generation of a view of a three-dimensional scene are

somewhat analogous to the processes involved in taking a photograph.

� To take a snapshot, we first need to position the camera at a particular point in

space.

� Then we need to decide on the camera orientation (in Fig).

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� Finally, when we snap the shutter, the scene is cropped to the size of the "window"

(aperture) of the camera, and light from the visible surfaces is projected onto the

camera film.

� Following figure shows the general processing steps for modeling and converting a

world-coordinate description of a scene to device coordinates.

� Once the scene has been modeled, world-coordinate positions are converted to

viewing coordinates.

� The viewing-coordinate system is used in graphics packages as a reference for

specifying the observer viewing position and the position of the projection plane,

which we can think of in analogy with the camera film plane.

� Next, projection operations are performed to convert the viewing-coordinate

description of the scene to coordinate positions on the projection plane, which will

then be mapped to the output device.

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� Objects outside the specified viewing limits are clipped h m further consideration,

and the remaining objects are processed through visible-surface identification and

surface-rendering procedures to produce the display within the device viewport.

VIEWING COORDINATES

� Generating a view of an object in three dimensions is similar to photographing the

object.

� We can walk around and take its picture from any angle, at various distances, and

with varying camera orientations.

� Whatever appears in the viewfinder is projected onto the flat film surface.

� The type and size of the camera lens determines which parts of the scene appear in

the final picture.

� These ideas are incorporated into three dimensional graphics packages so that

views of

a scene can be generated, given the spatial position, orientation, and aperture size

of the "camera".

� To obtain a series of views of a scene, we can keep the view reference point fixed

and change the direction of N, as shown in Fig.

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� This corresponds to generating views as we move around the viewing-coordinate

origin.

� In interactive applications, the normal vector N is the viewing parameter that is

most often changed.

� By changing only the direction of N, we can view a scene from any direction

except along the line of V.

� To obtain either of the two possible views along the line of V, we would need to

change the direction of V.

Transformation from World to Viewing Coordinates

1. Translate the view reference point to the origin of the world-coordinate system.

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2. Apply rotations to align the xv, yv, and zv axes with the world xw, yw, and zw axes,

respectively.

� If the view reference point is specified at world position (xo yo, zo), this point is

translated to the world origin with the matrix transformation

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VISIBLE SURFACE DETECTION

� A major consideration in the generation of realistic graphics displays is identifying

those parts of a scene that are visible from a chosen viewing position.

� There are many approaches we can take to solve this problem, and numerous

algorithms have used for different types of applications.

� Some methods require more memory, some involve more processing time, and

some apply only to special types of objects.

� The various algorithms are referred to as visible-surface detection methods.

� Sometimes these methods are also referred to as hidden-surface elimination

methods.

1. Back-face detection

2. Depth-buffer method

3. A-buffer method

4. Scan-line method

5. Depth-sorting method

6. BSP-tree method

7. Area-subdivision b1ethod

8. Octree methods

9. Ray-casting method

10. Curved surfaces

11. wireframe methods

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BACK-FACE DETECTION

� A fast and simple object-space method for identifying the back faces of a

polyhedron is based on the "inside-outside" tests.

� A point (x, y, z) is "inside" a polygon surface with plane parameters A, B, C, and

D if

� When an inside point is along the line of sight to the surface, the polygon must be

a back face (we are inside that face and cannot see the front of it from our viewing

position).

� We can simplify this test by considering the normal vector N to a polygon surface,

which has Cartesian components (A, B, C).

� In general, if V is a vector in the viewing direction from the eye (or "camera")

position, as shown in Fig,

� then this polygon is a back face if

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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.

� Since object depth is usually measured from the view plane along the z axis of a

viewing system.

� Each surface of a scene is processed separately, one point at a time across the

surface.

� The method is usually applied to scenes containing only polygon surfaces, because

depth values can be computed very quickly and the method is easy to implement.

� But the method can be applied to non planar surfaces.

� With object descriptions converted to projection coordinates, each (x, y, z )

position on a polygon surface corresponds to the orthographic projection point (x,

y) on the view plane.

� Therefore, for each pixel position (x, y) on the view plane, object depths can be

compared by comparing z values.

� Following figure shows three surfaces at varying distances along the orthographic

projection line from position (x, y) in a view plane taken as the xv, yv plane.

� Surface S1, is closest at this position, so its surface intensity value at (x, y) is saved

� As implied by the name of this method, two buffer areas are required.

� A depth buffer is used to store depth values for each (x, y) position as surfaces are

� processed, and the refresh buffer stores the intensity values for each position.

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� Initially, all positions in the depth buffer are set to 0 (minimum depth), and the

refresh

� buffer is initialized to the background intensity.

� Each surface listed in the polygon tables is then processed, one scan line at a time,

calculating the depth (z value) at each (x, y) pixel position.

� The calculated depth is compared to the value previously stored in the depth buffer

at that position.

� If the calculated depth is p a t e r than the value stored in the depth buffer, the new

depth value is stored, and the surface intensity at that position is determined and in

the same xy location in the refresh buffer.

� We summarize the steps of a depth-buffer algorithm as follows:

126 PRABHU.S


A-Buffer

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

� The A- buffer method represents an antialiased, area-averaged, accumulation-

buffer method.

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

at each pixel position.

� In other words, it deals only with opaque surfaces and cannot accumulate intensity

values for more than one surface, as is necessary if transparent surfaces are to be

displayed .

� The A-buffer method expands the depth buffer so that each position in the buffer

can reference a linked list of surfaces.

� Thus, more than one surface intensity can be taken into consideration at each pixel

position, and object edges can be ant aliased.

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UNIT III

COLOR MODELS

� A color model is a method for explaining the properties or behavior of color within

some particular context.

Chromaticity diagram:

� Chromaticity diagram is a convenient space coordinator representation of all the

colors and the mixture of colors.

How colors are represented here:

� The various colors are represented along the perimeter of the curve.

� The corner representing the 3 primary colors.

Uses of Chromaticity Diagram

� Comparing color gamuts for different sets of primaries.

� Identifying complementary colors.

� Determining dominant wavelength and purity of a given color.

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Hue

� This is the predominant spectral color of the received light.

� The color itself is its hue or tint.

� Green leaves have a green hue, red apple has a red hue.

Saturations:

� This is the spectral purity of the color light.

� Saturated colors are vivid, intense, deep

RGB COLOR MODEL

� In this color model, the three primaries Red, Green and Blue are used.

� Here color is expressed as,

C˄ = RR + CG + BB

� We can represent this model in unit cube as shown in following figure,

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� The origin represents black

� The vertex with coordinates (1, 1, 1) is white.

� The magenta vertex is obtained by adding red and blue.

� The yellow vertex is obtained by adding green and red and so on.

Additive model

� Intensities of the primary colors are added to produce other colors.

� Each color point within the bounds of the cube can be represented as the triple

(R, G, B)

� Where values for R, G, and B are assigned in the range from 0 to 1.

� Shades of gray are represented along the main diagonal of the cube from the

origin (black) to the white vertex.

YIQ COLOR MODEL

� In the YIQ color model, luminance (brightness) information is contained in the Y

parameter, while chromaticity information (hue and purity) is incorporated into the

1 and Q parameters.

� A combination of red, green, and blue intensities are chosen for the Y parameter to

yield the standard luminosity curve.

� Since Y contains the luminance information, black-and-white television monitors

use only the Y signal.

� The largest bandwidth in the NTSC video signal (about 4 MHz) is assigned to the

Y information.

� Parameter I contains orange-cyan hue information that provides the flesh-tone

shading, and occupies a bandwidth of approximately 1.5 MHz.

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� Parameter Q carries green-magenta hue information in a bandwidth of about 0.6

MHz.

� An RGB signal can be converted to a television signal using an NTSC encoder,

which converts RGB values to YIQ values.

� Then modulates and superimposes the I and Q information on the Y signal.

� The conversion from RGB values to YIQ values is accomplished with the

transformation.

NTSC signals

� An NTSC video signal can be converted to an RGB signal using an NTSC

decoder.

� Which separates the video signal into the YIQ components, then converts to RGB

values.

� We convert from YIQ space to RGB space with the inverse matrix transformation

RGB into YIQ

� An RGB signal can be converted to a television signal using an NTSC encoder

which converts RGB values to YIQ values.

� This conversion from RGB values to YIQ values is accomplished with the

transformation.

YIQ into RGB

� An NTSC video signal can be converted to an RGB signal using an NTSC

decoder.

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� Which separates the video signal into the YIQ components, then converts to

RGB values.

� We can convert from YIQ space to RGB space with the inverse matrix

transformation.

CMY COLOR MODEL

� In this model cyan, magenta, and yellow (CMY) are used as a primary colors.

� This model is useful for describing color output to hard-copy devices.

Video Monitors Vs Printers, Plotters:

� Video monitors produce a color pattern by combining light from the screen

phosphors.

� Whereas, hard-copy devices such as plotters produce a color picture by coating a

paper with color pigments.

Subtractive process

� It is a subtractive process.

� As we have noted, cyan can be formed by adding green and blue light.

� Therefore, when white light is reflected from cyan-colored ink, the reflected light

must have no red component.

� That is, red light is absorbed, or subtracted, by the ink.

� Similarly, magenta ink subtracts the green component from incident light, and

yellow subtracts the blue component.

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� A unit cube representation for the CMY model is illustrated in Fig.

� In the CMY model, point (1, 1, 1) represents black,

� Because all components of the incident light are subtracted.

� The origin represents white light.

� Equal amounts of each of the primary colors produce grays, along the main

diagonal of the cube.

� A combination of cyan and magenta ink produces blue light.

� Because the red and green components of the incident light are absorbed.

� Other color combinations are obtained by a similar subtractive process.

Printing Process

� The printing process often used with the CMY model generates a color point with

a collection of four ink dots, (like RGB monitor uses a collection of three phosphor

dots).

• Three dots are used for each of the primary colors (cyan, magenta, and

yellow).

• And one dot is black.

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� A black dot is included because the combination of cyan, magenta, and yellow

inks typically produce dark gray instead of black.

Conversion of RGB into CMY

� We can express the conversion from an RGB representation to a CMY

representation

with the matrix transformation

� Where the white is represented in the RGB system as the unit column vector.

Conversion of CMY into RGB

� Similarly, we convert from a CMY color representation to an RGB representation


� Where black is represented In the CMY system as the unit column vector.

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HSV COLOR MODEL

� Instead of a set of color primaries, the HSV model uses color descriptions that

have a more intuitive appeal to a user.

� To give a color specification, a user selects a spectral color and the amounts of

white and black that are to be added to obtain different shades, tints, and tones.

� Color parameters in this model are hue ( H ), saturation(S), and value (V).

� The three-dimensional representation of the HSV model is derived from the RGB

cube.

� If we imagine viewing the cube along the diagonal from the white vertex to the

origin (black).

� We see an outline of the cube that has the hexagon shape shown in Fig.

� The boundary of the hexagon represents the various hues, and it is used as the top

of the HSV hexcone.

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� In the hexcone, saturation is measured along a horizontal axis.

� Value is along a vertical axis through the center of the hexcone.

� Hue is represented as an angle about the vertical axis, ranging from 0" at red

through 360".

� Vertices of the hexcone are separated by 60" intervals.

� Yellow is at 600, green at 120

0, and cyan opposite red at H = 180

0.

� Complementary colors are 1800 apart.

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ANIMATION

� Computer animation generally refers to any time sequence of visual changes in a

scene.

� In addition to changing object position with translations or rotations, a computer-

generated animation could display time variations in object size, color,

transparency, or surface texture.

� Computer animations can also be generated by changing camera parameters, such

as position, orientation, and focal length.

� And we can produce computer animations by changing lighting effects or other

parameters and procedures associated with illumination and rendering.

DESIGN OF ANIMATION SEQUENCES

� In general, an animation sequence is designed with the following steps:

1. Storyboard layout

2. Object definitions

3. Key-frame specifications

4. Generation of in-between frames

Storyboard Layout

� The storyboard is an outline of the action.

� It defines the motion sequence as a set of basic events that are to take place.

� Depending on the type of animation to be produced, the storyboard could consist

of a set of rough sketches or it could be a list of the basic ideas for the motion.

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Object Definition

� An object definition is given for each participant in the action.

� Objects can be defined in terms of basic shapes, such as polygons or splines.

� In addition, the associated movements for each object are specified along with the

shape.

Keyframe

� A keyframe is a detailed drawing of the scene at a certain time in the animation

sequence.

� Within each key frame, each object is positioned according to the time for that

frame.

� Some key frames are chosen at extreme positions in the action.

� Others are spaced so that the time interval between key frames is not too great.

� More key frames are specified for intricate motions than for simple, slowly

varying motions.

Generation of in-between frames

� In-betweens are the intermediate frames between the key frames.

� The number of in-betweens needed is determined by the media to be used to

display the animation.

� Film requires 24 frames per second, and graphics terminals are refreshed at the rate

of 30 to 60 frames per second.

� Typically, time intervals for the motion are set up so that there are from three to

five in-betweens for each pair of key frames.

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RASTER ANIMATIONS � On raster systems, we can generate real-time animation in limited applications

using raster operations.

� Two dimensional rotations in multiples of 90" are also simple to perform,

although we can rotate rectangular blocks of pixels through arbitrary angles using

antialiasing procedures.

� To rotate a block of pixels, we need to determine the percent of area coverage for

those pixels that overlap the rotated block.

� Sequences of raster operations can be executed to produce real-time animation of

either two-dimensional or three-dimensional objects, as long as we restrict the

animation to motions in the projection plane.

� Then no viewing or visible surface algorithms need be invoked.

� We can also animate objects along two-dimensional motion paths using the

color -table transformations.

� Here we predefine the object at successive positions along the motion path, and set

the successive blocks of pixel values to color-table entries.

� We set the pixels at the first position of the object to "on" values, and we set the

pixels at the other object positions to the background color.

� The animation is then accomplished by changing the color-table values so that the

object is "on" at successively positions along the animation path as the preceding

position is set to the background intensity (Fig).

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KEY-FRAME SYSTEMS

� We generate each set of in-betweens from the specification of two (or more) key

frames.

� Motion paths can be given with a kinematic dm-ripti011 as a set of spline curves,

or the motions can be physically based by specifying the force acting on the

objects to be animated.

� For complex scenes, we can separate the frames into individual components or

objects called cels (celluloid transparencies), an acronym from cartoon animation.

� Given the animation paths, we can interpolate the positions of individual objects

between any two times.

� With complex object transformations, the shapes of objects may change over time.

� Examples are clothes, facial features, magnified detail, evolving shapes, exploding

or disintegrating objects, and transforming one object into another object.

� If all surfaces are described with polygon meshes, then the number of edges per

polygon can change from one frame to the next.

� Thus, the total number of line segments can be different in different frames.

MORPHING

� Transformation of object shapes from one form to another is called morphing,

� Which is a shortened form of metamorphosis.

� Morphing methods can he applied to any motion or transition involving a change

in shape.

� Given two key frames for an object transformation, we first adjust the object

specification in one of the frames so that the number of polygon edges (or the

number of vertices) is the same for the two frames.

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� This preprocessing step is illustrated in Fig.

� A straight-line segment in key frame k 15 transformed into

� Two line segments in kev frame k t 1. Since key frame k + 1 has an extra vertex,

� n'e add n veytex bctr\.rtw \wtices 1 and 2 in kcv frame k to balance the number of

� vertices (and edges) In the two key frames. Using linear interpolation to generate

� the in-betweens. wc trmsition the added vertex in key frclme k into vertex 3'

� along the straight-linv path shown in Fis. 16-7. An eianlple ol a triangle linearly

� cxp"11ding into ,I quad~.~lateral is given In Fig. 16-8. Figures 16-9 and 16-10

show

� examples uf morphing 111 television advertising

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OPENGL

(Open Graphics Library)

OpenGL is the precccmier environment for developing portable, interacting 2D

and 3D Graphics applications.

Advantages:

� OPENGL is truly open, vendor, neutral, multiplatform graphics standard.

� Stable.

� Reliable and portable

� Scalable

� Easy to use.

� Well documented.

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Features:

� It supports 3D transformation.

� It supports different color model

� It supports lighting (flat shading, Gouraud shading, Pong shading)

� It supports rendering.

� It supports different modeling

� It supports other special effects (atmosphere form, α-blending, motion blur)

OPNGL OPERATION:

GLUT: (OpenGL utility took kit);

� It is a window system independent to build for writing OPENGL programs.

� It implements a simple windowing application programming interface. [API]

for OPENGL.

� GLUT provides a portable API as one can write a single OPENGL program that

works across all PC and workstation OS platforms.

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Sample Program:

Void main (int argc, char ** argc)

{

glutInit (&argc, argv);

glutInitDisplay Mode (Glut-Single Glut-RGB);

glutInitWindowsize (640, 480);

glutInitWindowPosition (100, 150);

glutCreateWindow (“my first attempt”);

glutDisplayFunc (myDisplay);

glutReshapeFunc (myReshape);

glutMouseFunc (myMouse);

glutKeyboardFunc (myKeyboard);

myInit ();

glut mainloop ();

}

� glutInit (&argc, argv):

� It initializes the OPENGL Utility Toolkit. Its arguments are the standard

ones for parsing information about the command line.

� glutInitDisplay Mode (Glut-Single Glut-RGB):

� This function specifies how the display should be initialized.

� The argument indicates a single displayed buffer with RGB color model.

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� glutInitWindowsize (640, 480):

� This function specifies that this screen window should initially be 640

pixels wide by 480 pixels height.

� glutInitWindowPosition (100, 150):

� This function specifies that the windows upper left corner should be

positioned on the screen 100 pixels over from the left edge and 150

pixels down from the top.

� glutCreateWindow (“my first attempt”):

� This function actually opens and displays the screen window, putting

the title “my first attempt”.

� glutDisplayFunc (myDisplay):

� Whenever the system determines that a window should be redrawn on

the screen.

� glutReshapeFunc (myReshape):

� Screen windows can be reshaped by the user, usually by dragging a

corner of the window to a new position with the mouse.

� glutMouseFunc (myMouse):

� When one of the mouse button is pressed or released a mouse event as

issued.

� My mouse is registered as a function to be called when a mouse event

occurs.

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� glutKeyboardFunc (myKeyboard):

� This command register the function myKeyboard(): with a event of

pressing or releasing some key on the keyboard.

BASIC GRAPHICS PRIMITIVES

� OPENGL provide tools for drawing all of the output primitives such as points,

lines, polygons, and polylines.

� They are defined by one or more vertices.

� To draw objects in OPENGL, you pass it a list of vertices.

� The list occurs between the two OPENGL function calls glBegin () and glEnd ().

glBegin (GL_POINTS);

glVertex 2i (100, 50);

glVertex 2i (100, 130);

glVertex 2i (150, 130);

glEnd ();

Format of OPENGL commands:

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OPENGL Data Types:

Suffix Datatype Typical

Char++Type

OPENGL TypeName

B 8 bit integer Signed char GLbyte

S 16 bit integer short GLshort

I 32 bit integer int or long Glint, GLsize i

F 32 bit floating point float GLfloat, GLclamp F

D 64 bit floating point double Gldouble, GLclamped

Ub 8 bit unsigned number unsigned char GLUbyte, GLUdean

Us 16 bit unsigned number unsigned short GLUshort

Ui 32 bit unsigned number unsigned int or

unsigned long

GLUnit, GLenum,

GLbitfield.

� The size of a point can be set with glpointsize() which takes one floating point

argument. The color of a drawing can be specified using

Glcolor3f(red,green,blue);

� Where the values of red, green and blue vary between 0.0 and 1.0.

� To draw a line between (40, 100) and (202, 9) we use

glBegin (GL_lines);

givertex 2i (40, 100);

glvertex 2i (202, 96);

glEnd ();

� Polyline is a collection of line segment joining end to end.

� In OPENGL, a polyline is called a line strip and is drawn by specifying the

vertices in turn, between glBegin (GL_LINE_STRIP) and glEnd ().

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glBegin (GL_LINE_STRIP)

glvertex 2i (20, 10)




glEnd ();

glFlush ();

� Line can be drawn using moveto () and lineto () also.

� To drawn and aligned rectangle:

glRecti(Glint x1,Glint y1, Glint x2, Glint y2);

Other Graphics Primitives in OPENGL:

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GL_TRIANGLES: Takes the listed vertices three at a time and draws a separate

triangle for each.

GL_QUADS: Takes the vertices four at a time and draws a separate quadrilateral for

each.

GL_TRIANGLES_STRIP: Draws a series of triangles based as tripe of vertices: V0,

V1, V2 then V2, V1, V3 then V2, V3, V4

GL_TRIANGLES_FAN: Draws a series of connected triangles based on tripes of

vertices: V0, V1, V2 then V0, V2, V3 then V0, V3, V4

GL_QUAD_STRIP: Draws a series of quadrilaterals based on four somes of vertices:

first V0, V1, V3, V2 then V2, V3, V4, V5 then V4, V5, V7, V6, etc.

Example:

� The following code fragment specifies a 3D polygon to be drawn, in this case a

simple square.

� Note that in this case the same square could have been drawn using the

GL_QUADS and GL_QUAD_STRIP primitives.

GLfloat p1[3] = {0, 0, 1};

GLfloat p2[3] = {1, 0, 1};

GLfloat p3[3] = {1, 1, 1};

GLfloat p4[3] = {0, 1, 1};

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glBegin (GL_POLYGON);

glvertex3fv (p1);

glvertex3fv (p2);

glvertex3fv (p3);

glvertex3fv (p4);

glEnd ();

DRAWING 3D SCENES WITH OPENGL

Viewing process & Graphics Pipeline

� All of our 2D drawing so far has actually used a special case of 3D viewing, based

on a simple “parallel projection:.

� We have been use the “camera” as shown below,

� The “eye” that is viewing the scene looks along the z-axis at the “window”

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� The view volume of the camera is a rectangular parallel piped.

� Whose 4 side walls are determined by the border of the window.

� Other two walls are determined by a near plane and far plane.

� Points lying inside the view volume are projected onto the window along lines

parallel to the z-axis.

� Ignore the 2 component of those points, so that the 3D point (x1,y1,z1) projected to

(x1,y1,0).

� Points lying outside the view volume are clipped off.

� A separate view port transformation maps the projected points from the window to

viewport on display device.

� Following figure show a camera immersed in a scene.

� The scene consist of a block

� The image produced by the camera is also shown.

� The graphics pipeline implemented by OpenGL does its major work through

matrix transformations

� The important three matrices are

i. Model view matrix

ii. Projection matrix

iii. Viewport matrix

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Model View matrix:

� It basically provides what we have calling the CT,

� It combines two effects.

• sequence of modelling transformations applied to objects

• transformation that orients and positions the camera in space.

� The model view matrix is a single matrix in the actual pipeline.

� The modelling matrix is applied and then the viewing matrix.

� So the model matrix is in fact the produced VM.

VM = Viewing matrix.

M = modelling matrix.

Projection matrix:

� It scales and shifts each vertex n a particular way.

� So that all those vertices that inside the view volume will inside a standard cube.

� The projection matrix effectively squashes the view volume into the cube centred

at the origin.

� The projection matrix also reverse the sense of the z-axis.

� So that increasing values of z, increasing values of depth of a point from the eye.

� The following figure shows how the block is transformed into a different block.

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� Clipping is now performed, which eliminates the portion of the block that lies

outside the standard cube.

Viewport matrix:

� Finally viewport matrix maps the surviving portion of the block into a “3D

viewport”.

� This matrix maps the standard cube into a block shape.

� Whose x & y values extend across the viewport.

� Whose 2 component extends from 0 to 1

� That can be described in following figure.

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DRAWING THREE DIMENSIONAL OBJECTS

1. 3D Viewing Pipeline

� The world co-ordinate selected for display is called a window.

� An area on a display device to which a window is mapped is called viewport.

� The window defines what is to be viewed; the viewport defines where it is to be

displayed.

� Viewports are typically defined within the unit square.

� This provides a means for separating the viewing and other transformations from

specific output-device requirements.

� So that the graphics packages is largely device-independent.

OPENGL functions for setting up transformations

Modeling Transformation (Model View Matrix) → glTranslatef ()

→ glRotatef ()

→ glScalef ()

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Viewing Transformation (Model View Matrix) → gluLookAt ()

Projection Transformation (Model View Matrix) → glFrustum ()

→ gluPerspective ()

→ glortho ()

→ gluOrtho2D ()

Viewing Transformation → glViewport ()

To apply transformations in 2D case, OPENGL uses:

� glscaled (sx, sy, 1:0):

→ Postmultiply CT by a matrix that performs a scaling by sx in x and by

sy in y;

→ put the result back into CT (current transformation). No scaling in z

in done.

� glTranslated(dx, dy, 0):

→ Postmultiply CT by a matrix that performs a translation by dx in x

and by dy in y;

→ put the result back into CT (current transformation). No translation in

z in done.

� glRotated (angle, 0, 0, 1):

→ Postmultiply CT by a matrix that performs a rotation through angle

degrees about the z-axis. Put the result back into CT.

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→ To initialize the CT to the identity transformation OPENGL provides

glLoadIdentity().

3D Viewing – Model View Matrix

Code:

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

// Viewing Transform

gluLookAt (eyex, eyez, lookAt x, lookAt y, lookAt z, up X, up Y, up Z);

// Model Trnasform

glTrnaslatef (del x, del y, del z);

glRotatef (angle, i, j, k);

glScalef (mult x, mult y, mult z);

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UNIT IV

INTRODUCTION TO SHADING MODEL

� A shading model dictates how light is scattered or reflected from a surface.

� A shading model frequently used in graphics in two types of light sources.

• Point light sources

• Ambient light

� These light sources “shine” on the various surfaces of the objects.

� The incident light interact with the surface in three different ways.

• Some is absorbed by the surface and converted into heat.

• Some is reflected from the surface.

• Some is transmitted into the interior of the object, as in the case of piece of the

glass.

Black body

� If all the incident light is absorbed, the object appears black and is known as black

body.

� We focus on the part of the light that is reflected or scattered form the surface.

� Some amount of these reflected light travels and reach the eyes, causing the object

to be seen.

� There are two types of reflection of incident light.

• Diffuse scattering

• Specular reflection

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DIFFUSE SCATTERING

� It occurs when some of the incident light penetrates the surface slightly and is re-

radiated uniformly in all directions.

� Scattered light interacts strongly with the surface, so its color is usually affected by

the nature of the material out of which the surface is made.

Computing the diffuse component

� Suppose that light falls from a point source onto one side of the facet.

� A fraction of light is reradiated diffusely in all directions from that side.

� Some fraction of the reradiated part reaches the eye, with the intensity denoted by Id.

� The following figure(a) shows the cross section, a point source illuminating a facet S.

� In figure (b), the facet is turned partially away from the light source through an

angle θ.

Lambert’s Law

� The area subtended is now only the fraction cos (θ).

� So the brightness of S is reduced by that same fraction.

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� This relationship between brightness and surface orientation is often called

Lambert’s law.

� For the internsity of the diffuse component, we can adopt the expression.

Id = Ispd * s.m/ |s||m|

� Where Is is the intensity of the light source.

� pd is the diffuse reflection coefficient

� the following figure shows how the spheres appears when it reflects diffuse light,

for six different reflection coefficient.

SPECULAR REFLECTION

� Real objects do not scatter objects uniformly in all directions.

� So specular component is added to the shading model.

� Specular reflection causes highlights, which can add significantly to the realism of

a picture when objects are shiny.

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FLAT SHADING

� When a face is flat (like the roof of a barn) and the light sources are quite distant.

� The diffuse light component varies little over different points on the roof.

� In such cases it is reasonable to use the same color for every pixel covered by the

face.

� Flat shading is established in OpenGL by using the command.

glShadeModel(GL_FLAT);

� The following figure shows a buckyball and sphere rendered by means of flat

shading.

� The individual faces are clearly visible on both objects.

� Edges between faces actually appear more pronounced that they would be on an

actual physical object, due to the phenomenon in the eye known as lateral

inhibition.

� Specular highlights are rendered poorly with flat shading.

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� Because an entire face is filled with a color that was computed at only one vertex.

SMOOTH SHADING

� Smooth shading attempts to de-emphasize edges between faces by computing

colors at more points on each face.

� The two principle types of smooth shadings are

1. Gouraud Shading

2. Phong Shading

� OpenGL does only Gouraud shading.

GOURAUD SHADING

� Computationally speaking, Gouraud shading is modestly more expensive than flat

shading.

� Gouraud shading is established in OpenGL with the use of the function.

o glShadeModel(GL_SMOOTH);

� The following figure shows a buckyball and a sphere rendered by means of

Gouraud shading.

� The buckyball looks the same as when it was rendered with flat shading.

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� Because the same color is associated with each vertex of a face, so interpolation

changes nothing.

� But the sphere looks much smoother.

� The edges of the faces are replaced by a smoothly varying color across the object.

� The following figure suggests how Gouraud Shading reveals the “underlying”

surface approximated by the mesh.

� The polygonal surface’s shown in coss section, with vertices V1, V2 etc.

� The imaginary smooth surface is suggested as well

� Properly computed vertex normals m1, m2 etc, perpendicular to this imaginary

surface, so that normals of correct shading will be used.

� The color is then made to vary smoothly between vertices.

� Gouraud shading does not picture highlights well.

� Highlights are better reproduced by using Phong shading.

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PHONG SHADING

� Greater realism can be achieved.

� Particularly with regard to highlights on shiny objects.

� This is done by approximations of the normal vector to the face at each pixel.

� This type of shading is called Phong Shading.

� When computing Phong Shading, we find the normal vector at each point on the

face of the objects.

� And we apply the shading model there to find the color.

� We compute the normal vector at each pixel by interpolating the normal vectors at

the vertices of the polygon.

� Following figure shows a projected face, with normal vectors m1, m2, m3 and m4

indicated at the four vertices.

� For the scanline 1/s the vector mleft and mright are found by linear interpolation.

� For instance

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� This interpolated vector must be normalized to unit length before it is used in the

shading formula

� Once mleft and mright are known, they are interpolated to form a normal vector at

each x along the scan line.

� Following fig. Shows an object rendered by using Gouraud Shading and the same

object is rendered by using Phong Shading.

� In Phong Shading, the direction of normal vector varies smoothly from point to

point and more closely approximates.

� The production of specular highlights is much more faithful than with Gouraud

Shading.

� It produces more realistic rendering.

Drawback:

� Phong Shading is relatively slow speed.

� More computation is required per pixel.

� Phong shading can take six to eight times longer than Gouraud Shading.

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Why OpenGL is not setup to do Phong Shading?

� Because it applied the shading model once per vertex right after the modelview

transformation

� Normal vector information is not passed to the rendering stage following the

perspective transformation and division.

ADDING TEXTURE TO FACES:

� The realism of an image is greatly enhanced by adding surface texture to the

various faces of a mesh object.

� Basic function is

texture (s, t)

� This function produces a color or intensity value for each value of s and t

between 0 and 1.

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TYPES:

� There are numerous sources of textures.

� The most common textures are

• Bitmap textures

• Procedural texture

BITMAP TEXTURES:

� Textures are often formed from bit representation of images such as digitized

photo, clip art or image can be previously in some program.

TEXELS:

� Texels formed from bitmap consists of an array, say textr[c][r] of color values

often called Texels.

� If the array has c columns and r rows, the indices c and r varies from 0 to c-1

and 0 to r-1respectively.

PROCEDURAL TEXTURE:

� Alternatively we can define a texture by mathematical function or procedure

example, the following spherical shape

� It can be generated by the function

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float fakeshape (float s, float t)

{

float r = sqrt((s-0.5) * (s-0.5) + (t-0.5) * (t-0.5))

if (r<0.3)

return 1-4/0.3 //sphere intensity

else

return 0.2 //dark background

}

PASTING THE TEXTURE ON TO A FLAT SURFACE:

� Since texture space itself is flat, it is simplest to paste texture on to a flat

surface.

� Example,

To define a quadrilateral phase and to position a texture on it,

four texture co-ordinates and 4-3D points are passed to OPENGL function.

glBegin (GL_QUADS)

glTexCoord2f (0.0, 0.0);

glvertex3f (1.0, 2.5, 1.5);

glTexCoord2f (0.0, 0.6); glvertex3f (1.0, 3.7, 1.5);



glEnd

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MAPPING A SQUARE TO RECTANGLE:

� The above figure shows the common case in which the four corners of the

texture square are associated with 4-corners of a rectangle.

� Producing repeated structure

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� The above figure shows the use of texture co-ordinates that tile the texture,

making it repeat.

ADDING SHADOWS OF OBJECTS:

� Shadows make an image much more realistic, it shows how the objects are

positioned with respect to each other.

� Following figure shows cube and sphere with and without shadows.

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� Shadows are absent in figure A, so it is impossible to see how far above the

plane, the cube and the sphere are floating.

� Shadow in figure B, give useful hints as the positions of the objects.

� Generally a shadow conveys lot of information.

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� To compute the shape of the shadow is cost.

� In the above figure, the shape of the shadow is determined by the projections

of each of the phases of the box on the plane of the floor.

� This provides the key for drawing the shadows.

SHADOW BUFFER:

� Different methods for drawing shadow uses a variant of the depth buffer, that

performs the removal of hidden surfaces.

� In this method, an auxiliary second depth buffer called shadow buffer, is

employed for each light sources. This recovers lot of memory.

� The rendering of shadow is done by two stages.

i. Loading the buffer.

ii. Rendering the scene.

BUILDING A CAMERA IN A PROGRAM

� In order to have fine control ever camera movements, we create and manipulate

our own camera in a program.

� We create a “camera” class that knows how to do all the things a camera does.

� Doing this is very simple and the payoff is high.

� In a program, we create a “camera” object called, say “cam” and adjust it with

functions as following,

cam.set(eye,look,up); //initialize the camera

cam.slide(-1,0,-2); //slide the camera forward and to the left

cam.roll(30); //roll it through 300.

cam.yaw(20); //yaw it through 200.

.

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.

etc

.

.

� The following program shows the basic definition of the camera class.

Class camera

{

private:

point3 eye;

vector3 u,v,n;

double viewAngle, aspect, nearDist, farDist;

void setModeViewMatrix();

public:

camera();

void set(point3 eye, point3 look, vecotr3 up);

void roll(float angle);

void pitch(float angle);

void yaw(float angle);

void slide(float delu, float delv, float deln);

void setshape(float vAng, float asp, float nearD, float farD);

};

� Here point3 and vecotr3 are the basic data types.

� The utility routine setModelViewMatrix() communicates the model view matrix to

OpenGL

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� It is used only by member functions of the class and needs to be called after each

change is made to the camera’s position.

� The following program shows the possible implementation of this routine.

Void camera :: setModelViewMatrix(void)

{

float m[16];

vector3 eVec(eye.x, eye.y, eye.z);

m[0]=u.x; m[4]=u.y; m[8]=u.2; m[12]=-eVec.dot(u);

m[1]=v.x; m[5]=v.y; m[9]=v.2; m[13]=-eVec.dot(v);

m[2]=n.x; m[6]=n.y; m[10]=n.2; m[14]=-eVec.dot(v);

m[3]= 0; m[7]= 0; m[11]= 0; m[15]= 1.0;

glMatrixMode(GL_MODELVIEW);

glLoadMatrix(m);

i. It can “slid” in three dimensions

ii. It can be rotated about any of three co-ordinate axes.

Sliding the Camera:

� Sliding a camera means to move it along of its own axes.

� That is, in the u, v or n direction, without rotating it.

movement along “n” : forward or backward

movement along “u” : left or right

movement along “v” : up or down

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Rotating the Camera:

� We want to roll, pitch or yaw the camera.

� Each of these involves a rotation of the camera about one of its own axes.

� To roll the camera is to rotate it about its won n-axis.

� This means that both the directions u and v must be rotated, as shown in figure.

� We form two new axes u' and v' that lie in the same plane as u and v.

� The functions pictch() and yaw() are implemented in a similar fashion.

CREATING SHADED OBJECTS

� Shading is a process used in drawing for depicting levels of darkness on paper by

applying media more densely or with a darker shade for darker areas, and less

densely or with a lighter shade for lighter areas.

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� There are various techniques of shading including cross hatching where

perpendicular lines of varying closeness are drawn in a grid pattern to shade an

area.

� The closer the lines are together, the darker the area appears.

� Likewise, the farther apart the lines are, the lighter the area appears.

� Light patterns, such as objects having light and shaded areas, help when creating

the illusion of depth on paper.

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� Fly a camera through space looking at various polygonal mesh objects.

� It Include ambient diffuse and specular light components.

� Provide a keystroke that switches flat and smooth shading.

SHADING METHODS

1. Circulism:

→ This is a very popular shading method among artists.

→ The idea is to draw very tiny circles that overlap and intertwine.

→ Building up tone can be tedious but the results are worth it.

→ This shading method is great for rendering a realistic skin texture.

→ Use a light touch and build up tone.

2. Blended circulism:

→ Graphite is scribbled onto the paper just as in the last method.

→ Using a blending stump, the graphite is blended in small circular motions.

→ This shading method is also great for skin textures.

3. Dark Blacks:

→ If a user wants dark blacks, try using charcoal.

→ For a dark tone with apply the graphite to the paper.

→ Any time user dealing with dark tones and graphite, there will be a shine

that results.

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→ This happens because the tooth of the paper absorbs the graphite quickly

and there are extra layers left on top. Glare/shine is a reality when working

with graphite.

4. Loose Cross Hatching:

→ It is simple and effective.

→ It is very looking too.

→ The basic idea of crosshatching is to overlap lines.

→ Start by drawing a set of diagonal lines next to each other.

→ Then rotate the drawing 90 degrees and draw another set of diagonal lines

that overlap the first set.

→ This can be repeated numerous times to build up tone. Crosshatching can be

as tight or as loose.

5. Tight Cross Hatching:

→ Using the ideas from loose crosshatching, this shading method takes it a

little further.

→ Tone is built up through repetition and a soft touch.

→ This shading methods works really well for animal.

→ It’s not perfect and some paper tooth will show through.

6. Powder Shading:

→ Powder shading is a sketching shading method.

→ In this style, the stumping powder and paper stumps are used to draw a

picture.

→ The stumping powder is smooth and doesn’t have any shiny particles.

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→ The poster created with powder shading looks more beautiful than the

original. The paper to be used should have small grains on it so that the

powder remains on the paper.

RENDERING THE TEXTURE:

� Rendering in a face ‘F’ is similar to Gauraud shading proceeds across the face

fixed by pixel.

� For each pixel it must determine the corresponding texture-coordinates (s, t)

and set the pixel to the proper color.

� Following figure shows the camera taking a snapshot of a face F with texture

pasted on it and the rendering in progress.

� The scan line y is being filled from xleft → xright.

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� For each x, along this scan line it must compute the correct position on the

face (p(xs, ys)) from that obtain the correct position (s*, t*) within the texture.

� The following diagram show that incremental calculation

� Make one of the objects in the scene a flat planar surface, on which is seen

shadows of other objects.


For each x, along this scan line it must compute the correct position on the

)) from that obtain the correct position (s*, t*) within the texture.

The following diagram show that incremental calculation texture co

DRAWING SHADOWS

Make one of the objects in the scene a flat planar surface, on which is seen

178

For each x, along this scan line it must compute the correct position on the

)) from that obtain the correct position (s*, t*) within the texture.

texture co-ordinates.

Make one of the objects in the scene a flat planar surface, on which is seen

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� A simple way of drawing a drop shadow of a rectangular object is to draw a gray

or black area underneath and offset from the object.

� In general, a drop shadow is a copy in black or gray of the object, drawn in a

slightly different position. Realism may be increased by:

i. Darkening the colors of the pixels where the shadow casts instead of

making them gray. This can be done with alpha blending the shadow with

the area it is cast on.

ii. Softening the edges of the shadow. This can be done by adding Gaussian

blur to the shadow’s alpha channel before blending.

� Shadows are one of the most important visual cues that we have for understanding

the spatial relationships between objects.

� Unfortunately, even modern computer graphics technology has a difficult time

drawing realistic shadows at an interactive frame rate.

� One trick that you can use is to pre-render shadows and then apply them to the

scene as a textured polygon.

� This allows the creation of soft shadows and allows the computer to maintain a

high frame rate while drawing shadows.

Step 1: Activate and position the shadows

→ First, activate the shadows and position them using Sketch Up’s Shadows toolbar.

Step 2: Draw the Shadows Only

→ Next, to render the shadows without the geometry.

→ To do this, create two pages in Sketch Up.

→ Put the objects in the scene in a different layer than Layer0.

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→ So that toggles the visibility of the layer containing the objects.

→ Have the first page draw the shadows and show the layer containing the objects

in the scene.

→ Have the second page hide the layer containing the objects in the scene.

→ When the user moves from the first page to the second page, the objects will

disappear, leaving the shadows only.

Step 3: Draw the Shadows from Above

→ Next, position the camera to view the shadows from directly above so that we can

use the resulting image to draw the shadows onto a ground plane polygon.

Step 4: Soften the Shadows

→ The shadows that are rendered by Sketch Up always have hard edges.

→ In order to make the shadows look more realistic, we can soften the shadows

using software such as Photoshop or Gimp that includes an image blur tool.

→ When you create the shadow image, you can use the “alpha” channel of the

image to make portions of the image transparent.

Step 5: Create a new Shadow Material

→ Next is to create a new material that uses the soft shadow image from the

previous step as a texture.

→ If the image that we created in the previous step has an alpha channel, then the

alpha channel will be used to carve out transparent areas in the shadow

material.

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Step 6: Apply the material to a ground polygon

→ Last, create a ground polygon that underlies the objects in the scene and apply

the shadow material to it.

→ This will create a semi transparent polygon where dark patches are the shadow

areas. Since the shadows are pre-computed, you should turn off the Shadow

option in Sketch up.

→ In computer graphics, shading refers to the process of altering a color based on

its lights and its distance from lights to create a photorealistic effect.

→ Shading is performed during rendering process.

Shadow Mapping:

� Shadow mapping is just one of many different ways of producing shadows in our

graphics applications.

� Shadow mapping is an image space technique, working automatically with objects

created.

Advantages:

• No knowledge or processing of the scene geometry is required.

• Only a single texture is required to hold shadowing information for each light.

• Avoids the high fill requirement of shadow volumes.

Disadvantages:

• Aliasing, especially when using small shadow maps.

• The scene geometry must be rendered once per light in order to generate the

shadow map for a spotlight.

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UNIT V

FRACTALS & SELF SIMILARITY

Fractal:

� is a rough or fragmented geometric shape that can be split into parts.

� Each of which is a reduced copy of whole.

� Such a property is called Self Similarity.

Self Similarity:

� A self similar object is exactly or approximately similar to a part of itself.

� Self Similarity is a typical property of fractals.

� Computers are particularly good at repetition.

� They will do something again and again without complaint.

� Recursion often makes a difficult geometric task extremely simple.

� Among other things, it lets one decompose o refine shapes into ever smaller ones,

conceptually ad infinitum.

[ ad infinitum = infinity; continue forever, without limit ]

Self- Similar Curves:

� Many of the curves and pictures has the property called Self-Similar.

� Some curves are exactly self similar

� And some curves are Statistically Self Similar

Exactly Self Similar:

� If a region is enlarged, the enlargement looks exactly like the original.

Statistically Self Similar:

� The wiggles and irregularities in the curve are the same “on the average”.

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Example:

� Nature provides examples that mimic statistical self similarity.

� The classic example is coast line.

Mandelbrot:

� The mathematician “Benit Mandelbrot” brought together and popularized investigations

into the nature and self-similarity.

� He called various forms of self-similar curves as fractals

� A line is one dimensional and plane is two dimensional.

� But there are “creatures” in between them.

� We shall define curves that are infinite in length yet lie inside a finite rectangle.

Stirred: being excited or provoked to the expression of an emotion.

Koch Curve:

� Very complex curves can be furnished recursively by repeatedly “refining” a simple

curve.

� The simplest example perhaps is the Koch curve.

� That is discovered by mathematician Helge Von Koch.

� This curve stirred great interest in the mathematical world because it produces an

infinitely long line within a region of finite area.

� Successive generation of Koch curves are denoted by K0, K1, K2........

TWO GENERATIONS OF THE KOCH CURVE

� The Zeroth generation shape K0 is just a horizontal line of length unity.

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� The curve K1 is shown in the above figure.

� To create K1, divide the line K0

� And replace the middle section with a triangular “bump” having sides of length 1/3.

� The total length of the line is evidently 4/3.

� The second order curve K2 is formed by building a bump on each of the four line

segments of K1.

� In this process, each segment is increased in length by a factor of 4/3.

� So the total length of the curve is 4/3 larger than that of the previous generation.

� Thus K1 has total length (4/3)i.

� Which increases as i increases.

� As i tends to infinity, the length of the curve becomes infinite.

Koch Snowflake:

THE FIRST FEW GENERATIONS OF THE KOCH SNOWLAKE

� It is formed out of three Koch Curves joined together.

� The perimeter of the ith

generation shape Si is the three times the length of a simple Koch

Curve, and S0 is 3(4/3)i

� The following figure shows third, fourth and fifth generation of Koch snowflake.

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KOCH SNOFLAKE, S3, S4, AND S5

PEANO CURVES (OR) SPACE-FILLING CURVES

� Peano curves are the fractal like structures that are drawn through a recursive process.

� Some of the curves shown in below, are space-filling curves or peano curves.

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� Such curves have a fractal dimension of 2.

� They completely fill a region of space.

Example:

� The two most famous peano curves are Hilbert and Sierpeniski curves.

� Some low order Hilbert curves are shown below

CREATING AN IMAGES ITERATED FUNCTIONS

� Another way to approach infinity is to apply a transformation to a picture again

and again and examine what results.

� This technique provides another fascinating way to create fractal shapes.

� This idea has been developed by “Bransley”, in which an image can be

represented by a handful of numbers.

Experimental Copier

� We take an initial image I0

� That can be put it through a special “photocopier” that produces a new image

as I1 shown in fig.

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Making New “Copies” From Old

� It is not just a simple copy of I0

� Rather it is a superposition of several reduced versions of I0.

� We then take I1 and feed it back into the copier again, to produce image I2.

� We repeat this process forever, obtaining a sequence of images I0, I1, I2--- called

the Orbit of I0.

Sierpinski Copier

� Consider a specific example of a copier that we might call the supercopier or S-

Copier.

� It superimposes three smaller versions of whatever image is fed into it.

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� That shows what one pass through S-Copier produces when the input is the

letter F.

� These three are smaller images could just as well overlap.

� The following fig. Shows the first few iterate that the S-Copier produces.

The First part of the orbit of I0 for the S-copier

� The figure suggests that the iterates converge to the Sierpinski triangle.

� At each iteration the individual component FS become one-half as large and they

triple in number.

� As more and more iterations are made, the FS approach dots in size, and these dots

are arranged in Sierpinski triangle.

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� The final image does not depend on the shape of the F at all, but only the nature of

the Super copier.

How the S-Copier make the images?

� It contains 3 lenses.

� Each of which reduces the input image to one-half its size.

� And move it to a new position.

� These three reduced and shifted images and superposed on the printed output.

� Scaling and shifting are easily done by affine transformations.

MANDELBROT SETS

� The Mandelbrot set is a mathematical set of points, whose boundary generates a

distinctive and easily recognisable two dimensional fractal shape.

� The set is closely related to the Julia Set.

� It generates similarly complex shapes.

� This is named after the mathematician Benoit Mandelbrot.

Iteration Theory

� Julia and Mandelbrot sets arises from a branch of analysis known as iteration

theory (or dynamical systems theory)

� This theory asks what happens when one iterates a function endlessly.

Mandelbrot Sets and Iterated Function Systems

� A view of the Mandelbrot Set is shown in following figure.

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� It is the black inner portion.

� It appears to consist of a cardioids along with a number of wart like circles glued

to it.

� In actuality, its border is astoundingly complicated.

� Its complexity can be explored by zooming in on a portion of the border and

computing a close-up view.

� In this theory the zooming can be repeated forever.

� The border is “infinitely complex”

� In fact, it is a fractal curve.

� Each point in the figure is shaded or colored according to the outcome of an

experiment run on an IFS.

� The IFS of internet is shown in fig.

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The Iterated function system for Julia and Mandelbrot sets

� It causes the particularly simple function

where C is some constant.

� That is the system produces each “output” by squaring its input and adding C.

� We assume that the process begins with the starting value S.

� So the system generates the sequence of values or Orbits.

d1 = (S)2 + C

d2 = ((S)2 + C)

2 + C

d3 = (((S)2 + C)

2 + C)

2 + C

d4 = ((((S)2 + C)

2 + C)

2 + C)

2 + C

� The orbit depends on two ingredients.

i. Starting point S

ii. Given value of C.

F(z) = Z2 + C

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JULIA SETS

� The Mandelbrot Set and Julia Sets are extremely complicated sets of points in the

complex plane.

� There is a different Julia Set denoted Jc, for each value of C.

� A close related variation is the Filled-in Julia Set denoted by KC.

� Which is easier to define.

� The Filled-in Julia Set KC

� Consider the iterated function system.

� Now we set to C is fixed chosen value.

� And examine what happens for different starting points.

Drawing Filled-in Julia Sets

� Process of drawing a Filled-in Julia Set is almost identical to that for the

Mandelbrot set.

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� We again choose a window in the complex plane and associate pixels with points

in the window.

� However, pixels correspond to different values of the different points.

� A single value of C is chosen, and then the orbit for each pixel position is

examined to see whether it explodes.

RANDOM FRACTALS

� Fractal shapes are completely deterministic.

� They are completely predictable (even though they are very complicated)

� In graphics the term fractal has become widely associated randomly generated

curves and surfaces that exhibit a degree of self similarity.

� These curves are used to produce “Naturalistic” shapes for representing objects

such as ragged mountains, grass and fire.

Fractalizing a segment

� The simplest random fractal is formed by recursively roughening or fractaling a lie

segment.

� At each step, each line segment is replaced with a random amount.

C

Random amount Replace S with this elbow

A S M B

L

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� The above figure shows this process applied to the line segment ‘S’ having the end

points A & B.

� S is replaced by two line segments A to C and from C to B.

� For fractal curve, point C is randomly chosen along the perpendicular bisector L of S.

Stages of Fractalization

� There are three stages of fractalization.

First Stage:

� The midpoint of AB is perturbed to form point C.

Note:- [Perturbed: in mathematical method, that give approximate solutions to

problems that cannot be solved exactly.]

Second Stage:

� Each of the two segments has its midpoint perturbed to form points D and E.

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Third Stage:

� At final stage, new points F...........I are added.

Calculation of fractalization in a program

C

Random amount Replace S with this elbow

A S M B

L

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� Line L passes through the midpoint M of segment S and is perpendicular to it.

� And point C along L has the parametric form

C(t) = M + (B-A)⊥ t

Where M = (A+B)/2

� The most fractal curves, t is modelled as a Gaussian random variable with mean

and some standard derivation.

� The following runtime fract() shown below

� That generates curves that approximation of actual fractals.

� This routine recursively replaces each segment in a random elbow.

fract (Point2 A, Point2 B, double stdDev)

{

//generate Fractal curve from A to B

double xDiff = A.x – B .x. yDiff = A.y – B.y;

Point2 C;

if(xDiff * XDiff + YDiff * yDiff < minLenSq)

cvs.lintTo(B.x, B.y);

else

{

stdDev *= factor;

double t = 0;

for (int i=0; i<12; i++)

t += rand()/32768.0;

t = (t-6) * stdDev;

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C.x = 0.5 * (A.x + B.x) – t * (B.y – A.y);

C.y = 0.5 * (A.y + B.y) + t * (B.x – A.x);

fract(A, C, stdDev);

fract(C, B, stdDev);

}

}

Drawing a Fractal Curve

double MinLenSq, factor; //global variables

void drawfractal(Point2 A, Point2 B)

{

double beta, stdDev;

factor = pow(2.0,(1.0-beta).2.0);

cvs.moveTo(A);

fract(A, B, stdDev);

}

OVERVIEW OF RAY TRACING

� Ray Tracing is a technique for generating an image by tracing the path of light

through pixels in an image plane.

� And simulating the effect of its encounters with virtual objects.

� This technique is used to produce a very high degree of visual realism.

� Usually higher than the “Scanline rendering”

� But greater computational cost.

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Introduction:

� Ray tracing (ray casting) provides related but even more powerful approach to

rendering scenes.

� The following figure gives the basic idea

� A buffer as a simple array of pixels positioned in space, with eye looking it into

the scene.

� The general question is what does the eye see through this pixel?

� A ray of light is arriving at eye through the pixel from some point p in the scene.

� The colour of the pixel is determined by the light emanates along the ray from

point p in the scene.

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Reverse Process

� In reality the process is revered.

� A ray is case form the eye through the centre of pixel and out into the scene.

� Its path is traced to see what object it hits first and at what point

� This process automatically solves the hidden surface problems.

� The first surface hit by the ray is the closest object to the eye.

� For the description of light source in the scene, the studying model is applied to the

point first hit.

� And components of light are computed.

� The resulting colour is then displayed in the pixel.

Features of Ray Tracing:

� Some of the interesting visual effects are easily incorporated

• Shadowing

• Reflection

• Refraction

� That provides dazzling realism that are difficult to create by any other method.

� It ability to work comfortably with richer geometric primitives such as

• Spheres

• Cones and

• Cylinders.

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OVERVIEW OF THE RAY-TRACING PROCESS

� The following code segment shows the basic step in ray tracer.

define the objects and light sources in the scene

setup the camera

for (int r = 0; r < nRows; r++)

for (int c = 0; c < nCols; c++)

{

1. Build the rc-th ray

2. Find all intersections of the rc-th ray with objects in the scene.

3. Identify the intersection that lies closest to, and in front of the ye

4. Compute the “hit point” where the ray hits this object, and the normal

vector at that point.

5. Find the color of the light returning to the eye along the ray from the

point of intersection

6. Place the color in the rc-th pixel.

}

� The scene to be traced by rays through geometric objects and light sources.

� A typical scene may contain

• Spheres,

• Cones,

• Boxes,

• Cylinders etc...

� The objects are described in some fashion stored as object in camera

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Computing hit point:

� When objects have been tested, the object with the smallest is the closest to the

location of the hit point on the object’s found.

Computing Color:

� The colour of the light that receiving by object, in the direction of the eye is

computed and stored in the pixel.

� The following figure shows simple scene consisting of some cylinders, spheres and

cones.

� The snow man consists, mainly spheres.

� Two light sources are also shown

Object List:

� Descriptions of all the objects are stored in an object list.

� This is a linked list of descriptive records as shown below.

� The ray that is shown intersects a sphere, cylinder and two cones.

� All the other objects are missed.

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� The object with the small hit time, a cylinder in the scene is identified.

→ The hit spot Phit is easily identified by the ray equation.

Phit = eye + dirr,c thit (hit spot)

INTERSECTING OF A RAY WITH AN OBJECT

� Consider the following code

Scene scn; //create a scene

Scn.read(“myScene.dat”); //read the SDL Scene file.

� Objects in the scene are created and placed in a list.

� Each object is an instance of generic shape such as sphere or cone.

� The following figure shows some o the generic shapes we shall be ray tracing.

Some common generic shapes used in ray tracing

� Implicit form of generic sphere is

F (x,y,z) = x2 + y

2 + z

2 – 1

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� For convenience use the notation F(p)

F (p) = |p|2 – 1

Generic Cylinder

F (x,y,z) = x2 + y

2 – 1 for 0<z<1

ADDING SURFACE TEXTURE

� Computer generated images can be made much more lively and realistic by

painting textures on various surfaces.

� The following figure shows a ray-traced scene with several examples of textures.

� OpenGL is used to render each face.

� For each face F, a pair of texture co-ordinates is attached to each vertex of face.

� Then openGL “painted” each pixel inside the face by using the colour of the

corresponding point within a texture image.

� Two principle types of texture can be used.

i. Solid texture

ii. Image texture

Solid Texture

� Solid texture is sometimes called “3D texture”.

� The object is considered to be carved out of a block of solid material that itself has

texturing.

� The ray tracer reveals the colour of the texture at each point on the surface of the

object.

� The texture is represented by a function texture (x,y,z) that produces an (r,g,b)

colour value at every point in space.

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Example:

� Imagine a 3D checker bound made up of alternating red and black cubes stacked

up throughout all of space.

� We position one of the “cubelets” with a vertex (0,0,0) and the size S=(S.x,S.y,S.x)

� All other cubes have his same size (a width of S.x, aheight of S.y etc)

� That are placed adjacent to one another in all three dimensions.

� It is easy to write an expression for such a checkerboard texture.

Jump(x,y,z) = ((int) (A+x/S.x) + (int) (A+y/S.y) + (int) (A+z/S.z))%2

� The following figure shows a generic sphere and generic cube “composed” of

material with this solid texture

Ray tracing of some objects with checkerboard solid texture

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� The colour of the material is the colour the texture

� Notice that the sphere and the cube are clearly made up of solid “cubelets”.

REFLECTIONS & TRANSPARENCY

� One of the great strength of the ray-tracing method is the ease with which it can

handle both reflection and refraction of light.

� This allow to build scenes of exquisite realism, containing

• Mirrors

• Fishbowls

• Lenses etc.,

� There can be multiple reflections in which light bounces off several shiny

surfaces before reaching the eye.

� These processes require the spawning and tracing of additional rays.

� The following figure shows a ray emanating from the eye,

→ In the direction dir

→ And hitting a surface at the point Ph.

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� When the surface is mirror like or transparent (or both) the light I that reaches the

eye may have five components

I = Iamb + Ispec + Irefl + Itran

where

Iamb : ambient component

Idiff : diffuse component

Ispe : specular component

Irefl : reflected light component arising from the light IR.

Itran : Transmitted light components, arising from the light IT.

� The first three are Familiar

• Ambient

• Diffuse

• Specular contributions

� The diffuse and specular parts arise light sources in the environment that are at Ph.

� The following figure shows how the number of contributions of light grows at

each contact point.

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� I is the sum of three components

• Reflected component R1

• Transmitted component T1

• Local component L1.

Local component:

� Is simply the sum of the usual ambient, diffuse and specular reflections at Pn.

� Local components depend only on actual light sources.

� They are not computed on the basis of casting secondary rays.

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� Figure (b) abstracts the various light components into a “tree” of light

contributions.

� The transmitted components arriving on the lift branches.

� The reflected components arriving on the right branches.

� At each node a local component must also be added, but for simplicity it is not

shown.

The refraction of light:

� When a ray of light strikes a transparent object, a portion of the ray penetrates the

object, as shown in fig.

� The ray will change direction from dir to t if the speed of light is different in

medium 1 than in medium 2.

� If the angle of incidence of the ray is � 1.

� Snell’s lay states that the angle of refraction � 2 will be

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� Where C1 is speed of light in medium 1.

� C2 is speed of light in medium 2.

� Only the ration C2/C1 is important.

� It is often called the Index of refraction of medium 2 with respect to medium 1.

� If � 1 equals zero, light hitting an interface at right angles in not bent.

BOOLEAN OPERATIONS ON OBJECTS

� According to CSG, complex shapes are defined by set operations (also called

Boolean operations) on simpler shapes.

� Objects such as lenses a hallow fishbowls are easily formed by combining the

generic shapes.

� Such objects are variously called compound objects (or) Boolean objects (or) CSG

objects.

� The ray tracing method extends in a very organized way to compound objects.

� It is one of the great strengths of ray tracing that it fits so naturally with CSG

models.

� We look at the examples of three Boolean operators.

• Union

• Intersection

• Difference

� The following shows the compound objects built from spheres.

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� Fig (a) is a lens shape constructed as the intersection of two spheres.

� That is a point is in the lens if and only if it lies in both spheres.

� Symbolically “L” is the intersection of the spheres S1 and S2, can be written as

L = S1 � S2

� Fig(b) shows a bowl, constructed using the difference operation.

� Applying the difference operation is analogous to removing material to cutting or

carving.

� The bowl is specified by

B = (S1 – S2) – C

� The solid globe S1 is “hollowed out” by removing all the points of the inner sphere S2.

� The top is then opened by removing all points in the cone C.

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UNION OF FOUR PRIMITIVES

� A point is in the union of two sets A and B denoted A�B, if it is in A or B or in

both.

� The following fig. Shows a rocket constructed as the union of two cones and two

cylinders.

� That is, R = C1 � C2 � C3 � C4

� Cone C1 rests on cylinder C2.

� Cone C3 is partially embedded in C2 and rests on the fatter cylinder C4.

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QUESTION BANK

UNIT I

2D PRIMITIVES

PART A

1.Define Output Primitives

� Graphics programming packages provide functions to describe a scene in terms of

these basic geometric structures, referred to as output primitives.

2.What are Simple geometric components?


pictures.

3.What are Additional output primitives?

� That can be used to construct a picture include

� circles and other conic sections,

� quadric surfaces,

� spline curves and surfaces,

� polygon color areas, and

� character strings.

4.Define Random-scan system or Vector System.

� It stores point-plotting instructions in the display list, and coordinate values in

these instructions are converted to deflection voltages that position the electron

beam at the screen locations to be plotted during each refresh cycle.

6.How the straight line is drawn in Analog Display Devices

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� 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.

� Linearly varying horizontal and vertical deflection voltages are generated that are

proportional to the required changes in the x and y directions to produce the

smooth line.

7.How the straight line is drawn in Digital Display Devices?

� Digital devices display a straight line segment by plotting discrete points between

the two endpoints.

� Discrete coordinate positions along the line path are calculated from the equation

of the line.

8.What is Stair step Effect (jaggies)?

� For a raster video display, the line color (intensity) is then loaded into the frame

buffer at the corresponding pixel coordinates.

� Reading from the frame buffer, the video controller then "plots" the screen pixels.

� Screen locations are referenced with integer values.

� So plotted positions may only approximate actual Line positions between two

specified endpoints.

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9.How the pixel positions are referenced?

� Pixcel positions referenced by scan line number and column number.

10.What is getpixel ( ) function?

� Sometimes we want to be able to retrieve the current frame buffer intensity setting

for a specified location.

� We accomplish this with the low-level function

getpixel (x, y )

11.What are Line Equations?

slope-intercept equation

y= m.x + b

where

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12.What are Circle Equations?

General form :

Circle Equation in polar form

Circle midpoint method equation

.

13.What are Ellipse equations?

General Ellipse equation

Ellipse equation in polar form

Ellipse midpoint method equation

14.Define Ellipse. Or Properties or Ellipse.

� An ellipse is an elongated circle.

� An ellipse is defined as the set of points such that the sum of the distances from

two fixted positions (foci) is the same for all points.

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15.What are Major and Minor axes in Ellipse?

Major Axes

� The major axis is the straight line segment extending from one side of the ellipse

to the other through the foci.

Minor Axes

� The minor axis spans the shorter dimension of the ellipse, bisecting the major axis

at the halfway position (ellipse center) between the two foci.

16.What is Attribute parameter?

� Any parameter that affects the way a primitive is to be displayed is referred to as

an attribute parameter.

� Some attribute parameters, such as

• color and

• size

17.What are the basic attributes of line?

� Basic attributes of a straight line segment are its

• type,

• its width, and

• its color.

18.What are the Line type attribute?

� Line-type attributes are

Solid Line

Dotted Line

Dashed Line

Dash-Dotted Line

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19.Define Direct storage scheme.

� With the direct storage scheme, whenever a particular color code is specified in an

application program, the corresponding binary value is placed in the frame buffer

for each-component pixel in the output primitives to be displayed in that color.

� A minimum number of colors can be provided in this scheme with 3 bits of storage

per pixel

20.Define Grayscale.

� With monitors that have no color capability, color functions can be used in an

application program to set the shades of gray, or grayscale, for displayed

primitives.

� Numeric values over the range from 0 to 1 can be used to specify grayscale levels,

which are then converted to appropriate binary codes for storage in the raster.

� This allows the intensity settings to be easily adapted to systems with differing

grayscale capabilities.

21.Tabulate the four level grayscale system

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22. What are Area-Fill Attributes?

� Options for filling a defined region include a choice between a solid color or a

patterned fill.

� These fill options can be applied to polygon regions or to areas defined with

curved boundaries.

� In addition, areas can be painted using various brush styles, colors, and

transparency parameters.

23.What are the types of fill styles?

Fill Styles

� Areas are displayed with three basic fill styles:

• hollow with a color border,

• filled with a solid color, or

• filled with a specified pattern or design.

24.What are Character Attributes?

� The appearance of displayed characters is controlled by attributes such as

• font,

• size,

• color, and orientation.

25.List out the styles of the characters.

� The characters in a selected font can also be displayed with assorted underlining

styles

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• Bold face

• Underline

• Italics

26.What are the transformations available in 2D?

� The basic geometric transformations are

• translation,

• rotation, and

• scaling.

� Other transformations that are often applied to objects include

• reflection and

• shear.

27.What is Translation?

� A translation is applied to an object by repositioning it along a straight-line path

from one coordinate location to another.

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28.What is 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 ө and the position (x1,y1) of the

rotation point (or pivot point) about which the object is to be rotated.

29.What is Scaling?

A scaling transformation alters the size of an object.

� This operation can be carried out for polygons by multiplying the coordinate

values (x, y) of each vertex by scaling factors sx, and sy, to produce the

transformed coordinates (x', y'):

30.What is differential scaling?

� When sx, and sy, are assigned the same value, a uniform scaling is produced that

maintains relative object proportions.

� Unique values for sx, and sy, result in a differential scaling.

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31.What is Reflection?

� A reflection is a transformation that produces a mirror image of an object.

� The mirror image for a two-dimensional reflection is generated relative to an axis

of reflection by rotating the object 180" about the reflection axis.

32.What is 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 a shear.

� Two common shearing transformations are those that shift coordinate x values and

those that shift y values.

PART B

1.Explain in detail Line Drawing algorithms with example.

2.Explain in detail Circle Drawing algorithms with example.

3.Explain in detail Ellipse Drawing algorithms .

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4.Exolain in detail about attributes of output primitives.

5.What are the 2D transformations available? Explain any two.

6. Exolain in detail about 2D viewing.

7.What is Line Clipping? Explain in detail Coheh-Southerland Line clipping.

8.What is polygon clipping? Explain in detail about Sutherland-Hodgeman Polygon

Clipping.

UNIT II

3D CONCEPTS

PART A

1.What is mean by Perspective?

Perspective : “The appearance of things relative to one another as determined by their

distance from the viewer”

2.What is Depth Cueing?

� Depth information is important so that we can easily identify, for a particular

viewing direction, which is the front and which is the back of displayed objects.

3.What are the types of Projections?

� There are two basic projection methods.

• Parallel Projection

• Perspective Projection

4.What is Parallel Projection?

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� In a parallel projection, coordinate positions are transformed to the view plane

along parallel Line.

A parallel projection preserves relative proportions of objects

5.What is Perspective Projection?



of projection).

� A perspective projection, on the other hand, produces realistic views but does not

preserve relative proportions.

6.What are the types of Parallel projections?

There are two types if parallel projections,

1. Orthographic parallel projection.

2. Oblique parallel projection

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7.Differenciate Parallel projection and Perspective projection.

Sn Parallel Projection Perspective Projection

1 It preserves relative proportions of

objects.

It does not preserves relative proportions

of objects.

2 It does not give us a realistic

representation 3D dimensional objects

It gives us a realistic representation 3D

dimensional objects

3 coordinate positions are transformed

to the view plane along parallel Line.

object positions are transformed to the

view plane along lines that converge to a

point called the projection reference

point

4. All objects are same size. Projections of distant objects are

smaller, and same size for closer

objects.

8.What is projection reference point?



of projection).

9.What are the types of 3D representations?

� Representation schemes for solid objects are often divided into two broad

categories,

3. Boundary representations

4. Space-partitioning representation

10.What are Boundary representations?

� Boundary representations (B-reps) describe a three-dimensional object as a set of

surfaces that separate the object interior from the environment.

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� Typical examples of boundary representations are polygon facets and spline

patches.

11. What is Space-partitioning representation?

� Space-partitioning representations are used to describe interior properties, by

partitioning the spatial region containing an object into a set of small, non

overlapping, contiguous solids (usually cubes).

� A common space-partitioning description for a three-dimensional object is an octree

representation.

12.What is Polygon Table?

� We specify a polygon surface with a set of vertex coordinates and associated

attribute parameters.

� An information for each polygon are placed into tables that are to be used in the

subsequent processing, display, and manipulation of the objects in a scene.

13.What are the types of Polygon tables?

� Polygon data tables can be organized into two groups:

3. geometric tables and

4. attribute tables.

14.What is Geometric table?

� It contain vertex coordinates and parameters to identify the spatial orientation of

the polygon surfaces.

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15.What is Attribute table?

� It includes parameters specifying the degree of transparency of the object and its

surface reflectivity and texture characteristics.

16.What are the lists created by the Geometric table?

� A convenient organization for storing geometric data is to create three lists:

4. a vertex table,

5. an edge table, and

6. a polygon table.

17.What is Polygon Mesh?

� Some graphics packages provide several polygon functions for modeling objects.

� A single plane surface can be specified with a function such as “fillArea”.

� But when object surfaces are to be tiled, it is more convenient to specify the

surface facets with a mesh function.

18.What is Triangle strip?

� One type of polygon mesh is the triangle strip.

� This function produces n - 2 connected triangles, .as shown in above figure.

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19.What is Quadrilateral mesh?

� Another similar function is the quadrilateral mesh.

� which generates a mesh of (n - 1) by (m - 1) quadrilaterals, given the coordinates

for an n by m array of vertices.

� Above figure shows 20 vertices forming a mesh of 12 quadrilaterals.

20.What is Spline?

� A spline is a flexible strip used to produce a smooth curve through a designated set

of points.

� Several small weights are distributed along the length of the strip to hold it in

position on the drafting table as the curve is drawn.

� The term spline curve originally referred to a curve drawn in this manner.

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21.What are the Spline specifications?

Spline Specifications

� There are three equivalent methods for specifying a particular spline representation:

4. We can state the set of boundary conditions that are imposed on the spline; or

5. We can state the matrix that characterizes the spline; or

6. We can state the set of blending functions (or basis functions) that determine

how specified geometric constraints on the curve are combined to calculate

positions along the curve path.

22.Give some examples of scalar quantities.

• energy,

• temperature,

• pressure,

• frequency.

23.Give some examples of vector quantities.

• velocity,

• force,

• electric fields,

• electric current.

24. Define Pseudo-color method.

� Pseudo-color methods are also used to distinguish different values in a scalar data

set, and color-coding techniques can be combined with graph and chart methods.

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� To color code a scalar data set, we choose a range of color and map the range of

data values to the color range.

� For example, blue could be assigned to the lowest scalar value, and red could be

assigned to the highest value.

25.What are the translations available in 3D?

The basic transformations are

1. Translation

2. Scaling

3. Rotation

And other tow transformations are

1. Shear

2. Reflection

26.What is 3D Translation?

� In a three-dimensional homogeneous coordinate representation, a point is

translated

from position P = (x, y, z) to position P' = (x', y', z') with the matrix operation

230 PRABHU.S


27.What is 3D 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.

28.What is 3D Shear?

� Shearing transformations can he used to modify object shapes.

� They are also useful in three-dimensional viewing for obtaining general projection

transformations.

� In two dimensions, we discussed transformations relative to the x or y axes to

produce

distortions in the shapes of objects.

� In three dimensions, we can also generate shears relative to the z axis.

29.What are Visible Surface Detection Methods?

A major consideration in the generation of realistic graphics displays is identifying

those parts of a scene that are visible from a chosen viewing position.

� Some methods require more memory, some involve more processing time, and

some apply only to special types of objects.

� The various algorithms are referred to as visible-surface detection methods.

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� Sometimes these methods are also referred to as hidden-surface elimination

methods.

30.What are the visible-surface detection methods? <write any 4>



14. A-buffer method

15. Scan-line method

16. Depth-sorting method

17. BSP-tree method

18. Area-subdivision b1ethod

19. Octree methods

20. Ray-casting method

21. Curved surfaces

22. wireframe methods

31.What is z-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.

32.What is A-Buffer method?

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

� The A- buffer method represents an antialiased, area-averaged, accumulation-

buffer method.

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PART B

1.Wht is projection? Explain it with types.

2.How to represent Polygon surfaces in 3D?

3.How to represent Curved surfaces in 3D?

4.How to visualize data sets?

5. What are the 3D transformations available? Explain any two.

6.What is viewing pipeline and viewing coordinate?

7.What are the types of visible-surface detection methods? Explain any two.

233 PRABHU.S


UNIT III

GRAPHICS PROGRAMMING

PART A

1.What is Color Model?

� A color model is a method for explaining the properties or behavior of color within some

particular context.

2.What are the uses of Chromaticity diagram?

� Comparing color gamuts for different sets of primaries.

� Identifying complementary colors.

� Determining dominant wavelength and purity of a given color.

3.Draw the RGB Unit cube.

4.Draw the CMY color model unit cube.

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5. What is mean by Subtractive Process?

� In CMY color model, cyan can be formed by adding green and blue light.

� Therefore, when white light is reflected from cyan-colored ink, the reflected light

must have no red component.

� That is, red light is absorbed, or subtracted, by the ink.

6. What are the dots used in Printing Proceses?

� The printing process often used with the CMY model generates a color point with a

collection of four ink dots, (like RGB monitor uses a collection of three phosphor dots).

• Three dots are used for each of the primary colors (cyan, magenta, and yellow).

• And one dot is black.

7. Why the block dot is included in Printing Process?

� A black dot is included because the combination of cyan, magenta, and yellow inks

typically produce dark gray instead of black.

8. How to convert RGB into CMY?

� We can express the conversion from an RGB representation to a CMY representation


235 PRABHU.S


� Where the white is represented in the RGB system as the unit column vector.

9. How to convert CMY into RGB?

� we convert from a CMY color representation to an RGB representation


� Where black is represented In the CMY system as the unit column vector.

10.Draw the HSV hexcone.

11. What is Computer Animation?

236 PRABHU.S


� Computer animation generally refers to any time sequence of visual changes in a

scene.

� In addition to changing object position with translations or rotations, a computer-

generated animation could display time variations in object size, color,

transparency, or surface texture.

12. What are the steps for designing animation sequences?

� In general, an animation sequence is designed with the following steps:





13. What is Storyboard Layout?

� The storyboard is an outline of the action.

� It defines the motion sequence as a set of basic events that are to take place.

� Depending on the type of animation to be produced, the storyboard could consist

of a set of rough sketches or it could be a list of the basic ideas for the motion.

14. Object Definition

� An object definition is given for each participant in the action.

� Objects can be defined in terms of basic shapes, such as polygons or splines.

� In addition, the associated movements for each object are specified along with the

shape.

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15. Keyframe

� A keyframe is a detailed drawing of the scene at a certain time in the animation

sequence.

� Within each key frame, each object is positioned according to the time for that

frame.

� Some key frames are chosen at extreme positions in the action.


� In-betweens are the intermediate frames between the key frames.

� The number of in-betweens needed is determined by the media to be used to

display the animation.

� Film requires 24 frames per second, and graphics terminals are refreshed at the rate

of 30 to 60 frames per second.

17.What is Morphing?

� Transformation of object shapes from one form to another is called morphing,

� Which is a shortened form of metamorphosis.

� Morphing methods can he applied to any motion or transition involving a change

in shape.

18. What are the matrices in Graphics pipeline of OpenGL?

� The important three matrices are

iv. Model view matrix

v. Projection matrix

vi. Viewport matrix

19.What is ModelView matrix?

� The model view matrix is a single matrix in the actual pipeline.

� It combines two effects.

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• sequence of modelling transformations applied to objects

• transformation that orients and positions the camera in space.

� the model matrix is in fact the produced VM.

VM = Viewing matrix.

M = modelling matrix.

20.What is Projection matrix?

� It scales and shifts each vertex n a particular way.

� So that all those vertices that inside the view volume will inside a standard cube.

� The projection matrix effectively squashes the view volume into the cube centred

at the origin.

� The projection matrix also reverse the sense of the z-axis.

21.What is Viewport matrix?

� viewport matrix maps the surviving portion of the block into a “3D viewport”.

� This matrix maps the standard cube into a block shape.

� Whose x & y values extend across the viewport.

� Whose 2 component extends from 0 to 1

22.List out the model view matrix transformation functions.

→ glTranslatef ()

→ glRotatef ()

→ glScalef ()

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23.List out the Projection matrix transformation functions.

→ glFrustum ()

→ gluPerspective ()

→ glortho ()

→ gluOrtho2D ()

24.What are the 2D transformation functions in OpenGL?

� glscaled (sx, sy, 1:0):

� glTranslated(dx, dy, 0):

� glRotated (angle, 0, 0, 1):

PART B

1.What is color model? Explain any two.

2.What is animation? Explain design animation sequences.

3.Explain in detail about OpenGL programming.

4.How basic graphics primitives are achieved in OpenGL?

5.How to draw 3D scenes in OpenGL?

6.How to draw 3D objecsts in OpenGL?

240 PRABHU.S


UNIT IV

RENDERING

UNIT IV

1.What is Shading Model?

� A shading model dictates how light is scattered or reflected from a surface.

� A shading model frequently used in graphics in two types of light sources.

• Point light sources

• Ambient light

2.How many ways the incident light interact with the surfaces?

� The incident light interact with the surface in three different ways.

• Some is absorbed by the surface and converted into heat.

• Some is reflected from the surface.

• Some is transmitted into the interior of the object, as in the case of piece of the

glass.

3.What is Black body?

� If all the incident light is absorbed, the object appears black and is known as black

body.

4.What are the types of reflection of incident light?

� There are two types of reflection of incident light.



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5.What is Diffuse Scattering?

� It occurs when some of the incident light penetrates the surface slightly and is re-

radiated uniformly in all directions.

� Scattered light interacts strongly with the surface, so its color is usually affected by

the nature of the material out of which the surface is made.

6.What is Lambert’s law?

� The area subtended is now only the fraction cos (θ).

� So the brightness of S is reduced by that same fraction.

� This relationship between brightness and surface irentation is often called

Lambert’s law.

7.What is diffuse reflection coefficient?

� For the internsity of the diffuse component, we can adopt the expression.

Id = Ispd * s.m/ |s||m|

� Where Is is the intensity of the light source.

� pd is the diffuse reflection coefficient

8.What is Specular reflection?

� Real objects do not scatter objects uniformly in all directions.

� So specular component is added to the shading model.

� Specular reflection causes highlights, which can add significantly to the realism of

a picture when objects are shiny.

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9.What are the commands used in OpenGL for shadings?

� Flat shading is established in OpenGL by using the command.

glShadeModel(GL_FLAT);


glShadeModel(GL_SMOOTH);

10.What are the types of Shading Models?

There are two main types

• Flat Shading

• Smooth Shading

Shading Model

Flat Shading Smooth Shading

Gouraud Shading Phong Shading

11.What is lateral inhibition?

� Edges between faces actually appear more pronounced that they would be on an

actual physical object, due to the phenomenon in the eye known as lateral

inhibition.

12.What is Gouraud Shading?

� Computationally speaking, Gouraud shading is modestly more expensive than flat

shading.


o glShadeModel(GL_SMOOTH);

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13. What is Phong Shading?

� Greater realism can be achieved.

� Particularly with regard to highlights on shiny objects.

� This is done by approximations of the normal vector to the face at each pixel.

� This type of shading is called Phong Shading.

14.What are the drawbacks of Phong shading?

� Phong Shading is relatively slow speed.

� More computation is required per pixel.

� Phong shading can take six to eight times longer than Gouraud Shading.

15.Why OpenGL is not setup to do Phong Shading?

� Because it applied the shading model once per vertex right after the modelview

transformation

� Normal vector information is not passed to the rendering stage following the

perspective transformation and division.

15.What is Texture?

A texture can be uniform, such as a brick wall, or irregular, such as wood grain or

marble.

� The realism of an image is greatly enhanced by adding surface texture to the

various faces of a mesh object.

16.What are the types of Texture?

� There are numerous sources of textures.

� The most common textures are

• Bitmap textures


244 PRABHU.S


17.What is the basic function used in Texture?

� Basic function is

texture (s, t)

� This function produces a color or intensity value for each value of s and t

between 0 and 1.

18.What is Bitmap textures?

� Textures are often formed from bit representation of images such as digitized

photo, clip art or image can be previously in some program.

19.What are Texels?

� Texels formed from bitmap consists of an array, say textr[c][r] of color values

often called Texels.

� If the array has c columns and r rows, the indices c and r varies from 0 to c-1

and 0 to r-1respectively.

20. What is Procedural Texture?

� Alternatively we can define a texture by mathematical function or procedure

example, the following spherical shape

245 PRABHU.S


21.Write a OpenGL program to generate the Procedural Texture.

float fakeshape (float s, float t)

{

float r = sqrt((s-0.5) * (s-0.5) + (t-0.5) * (t-0.5))

if (r<0.3)

return 1-4/0.3 //sphere intensity

else

return 0.2 //dark background

}

22.What are the advantages of adding shadows?

� Shadows make an image much more realistic.

� It shows how the objects are positioned with respect to each other.

� Using it we can identify the position of light source.

23. What is Shadow Buffer?

� Different methods for drawing shadow uses a variant of the depth buffer, that

performs the removal of hidden surfaces.

� In this method, an auxiliary second depth buffer called shadow buffer, is

employed for each light sources. This recovers lot of memory.

24.What is meant by Sliding the Camera?

� Sliding a camera means to move it along of its own axes.

� That is, in the u, v or n direction, without rotating it.

movement along “n” : forward or backward

movement along “u” : left or right

movement along “v” : up or down

246 PRABHU.S


25. What is meant by Rotating the Camera?

� We want to roll, pitch or yaw the camera.

� Each of these involves a rotation of the camera about one of its own axes.

� To roll the camera is to rotate it about its won n-axis.

26.What are the shading methods available?

7. Circulism:

8. Blended circulism:

9. Dark Blacks:

10. Loose Cross Hatching:

11. Tight Cross Hatching:

12. Powder Shading:

PART B

1. What is shading model? Explain it with types.

2. What is texture? How to add texture to faces?

3. Explain about the adding shadows to objects.

4. How to build a camera in your program?

5. What are the methods to create shaded objects?

6. Explain about drawing shadows.

247 PRABHU.S


UNIT V

FRACTALS

1.What is Fractal?

� is a rough or fragmented geometric shape that can be split into parts.

� Each of which is a reduced copy of whole.

� Such a property is called Self Similarity.

2.What is Self Similarity?

� Self Similarity is a typical property of fractals.

� A self similar object is exactly or approximately similar to a part of itself.

3.What is Koch Curve?

� Very complex curves can be furnished recursively by repeatedly “refining” a simple

curve.

� The simplest example perhaps is the Koch curve.

� That is discovered by mathematician Helge Von Koch.

� This curve stirred great interest in the mathematical world because it produces an

infinitely long line within a region of finite area.

4.What is Koch Snowflake?

� It is formed out of three Koch Curves joined together.

� The perimeter of the ith

generation shape Si is the three times the length of a simple Koch

Curve, and S0 is 3(4/3)i

5.What are peano curves?

Peano curves are the fractal like structures that are drawn through a recursive process.

248 PRABHU.S


6.What are the types of peano curves?

� The two most famous peano curves are Hilbert and Sierpeniski curves.

� Some low order Hilbert curves are shown below

7. How the S-Copier make the images?

� It contains 3 lenses.

� Each of which reduces the input image to one-half its size.

249 PRABHU.S


� And move it to a new position.

� These three reduced and shifted images and superposed on the printed output.

� Scaling and shifting are easily done by affine transformations.

8.What are Mandelbrot sets?

� The Mandelbrot set is a mathematical set of points, whose boundary generates a

distinctive and easily recognisable two dimensional fractal shape.

� The set is closely related to the Julia Set.

� It generates similarly complex shapes.

� This is named after the mathematician Benoit Mandelbrot.

9.What is Iteration Theory

� Julia and Mandelbrot sets arises from a branch of analysis known as iteration

theory (or dynamical systems theory)

� This theory asks what happens when one iterates a function endlessly.

10.What are Random Fractals?

� Fractal shapes are completely deterministic.

� They are completely predictable (even though they are very complicated)

� In graphics the term fractal has become widely associated randomly generated

curves and surfaces that exhibit a degree of self similarity.

� These curves are used to produce “Naturalistic” shapes for representing objects

such as ragged mountains, grass and fire.

11.What are the stages of Fractalization?

� There are three stages of fractalization.

250 PRABHU.S


First Stage:

� The midpoint of AB is perturbed to form point C.

Second Stage:

� Each of the two segments has its midpoint perturbed to form points D and E.

Third Stage:

� At final stage, new points F...........I are added.

12. How to Calculate fractalization in a program?

� Line L passes through the midpoint M of segment S and is perpendicular to it.

� And point C along L has the parametric form

C(t) = M + (B-A)⊥ t

Where M = (A+B)/2

� The most fractal curves, t is modelled as a Gaussian random variable with mean

and some standard derivation.

13. Write the program for drawing a Fractal Curve.

double MinLenSq, factor; //global variables

void drawfractal(Point2 A, Point2 B)

{

double beta, stdDev;

factor = pow(2.0,(1.0-beta).2.0);

cvs.moveTo(A);

fract(A, B, stdDev);

251 PRABHU.S


}

14.What is Ray Tracing?

� Ray Tracing is a technique for generating an image by tracing the path of light

through pixels in an image plane.

� And simulating the effect of its encounters with virtual objects.

� This technique is used to produce a very high degree of visual realism.

� Usually higher than the “Scanline rendering”

� But greater computational cost.

15.What are the features of Ray Tracing?

� Some of the interesting visual effects are easily incorporated

• Shadowing

• Reflection

• Refraction

16.How to compute hit point and color in Ray Tracing?

Computing hit point:

� When objects have been tested, the object with the smallest is the closest to the

location of the hit point on the object’s found.

Computing Color:

� The colour of the light that receiving by object, in the direction of the eye is

computed and stored in the pixel.

17.What is “object list” in Ray Tracing method?

252 PRABHU.S


� Descriptions of all the objects are stored in an object list.

� This is a linked list of descriptive records as shown below.

� The ray that is shown intersects a sphere, cylinder and two cones.

� All the other objects are missed.

18.What is Solid Texture?

� Solid texture is sometimes called “3D texture”.

� The object is considered to be carved out of a block of solid material that itself has

texturing.

� The ray tracer reveals the colour of the texture at each point on the surface of the

object.

19.What is called Compund Objects? Or

What is called Boolean Objects?

What is called CSG Objects?

� Complex shapes are defined by set operations (also called Boolean operations) on

simpler shapes.

� Objects such as lenses a hallow fishbowls are easily formed by combining the

generic shapes.

� Such objects are variously called compound objects (or) Boolean objects (or) CSG

objects.

253 PRABHU.S


20.What are the three Boolean operators?

• Union

• Intersection

• Difference

PART B

1. How create an image by iterated functions?

2. What are Mandelbrot sets and Julia sets?

3. Wxplain about Random fractals.

4. Explain in detail about ray tracing?

5. Explian about reflections and transparency.

6. How the Boolean operations are applied to objects?

254 PRABHU.S


ROAD MAP

UNIT I

Output primitives

→ Definition

→ Simple geometric components

→ Additional output primitives

Line

→ Stair step effect (jaggies)

Line-drawing algorithms

→ Slope-intercept equation

DDA algorithm

Bresenham algorithm

→ derivations

→ algorithm

→ problem

Circle algorithm

→ General form

→ Polar form

→ Midpoint circle algorithm

→ Theory & derivation

→ Algorithm

→ Problem

Ellipse algorithm

→ General form

→ Polar form

255 PRABHU.S


→ Midpoint ellipse algorithm

→ Theory & derivation

→ Algorithm

→ Problem

Attributes of output primitives

→ Attribute parameter

→ Line attributes

• Type,

o Solid lines,

o Dashed lines,

o And dotted lines

• Width

• color.

→ Pen and brush options

• Diagram

→ Line color

→ Curve attributes

→ Color and grayscale levels

→ Direct storage scheme

• Table 8 color code

→ Grayscale

• Table(4 level grayscale level)

→ Area-fill attributes

• Fill styles

→ Character attributes

• Font

256 PRABHU.S


• Size

• Color

• Diagram

2D transformation

→ Translation,

• Diagram

• Equation

• Matrix format

→ Rotation

• Diagram

• Equation

• Matrix format

→ Scaling

• Diagram

• Equation

• Differential scaling.

• Matrix format

→ Reflection

• Diagrams

• Definition

• Matrix format

→ Shear

• Diagram

• Equation

257 PRABHU.S


• Matrix format

Transformation functions

Translate (trans-atevector, matrixtranslate)

Rotate (theta, matrixrotate)

Scale (scalevector, matrixscale)

composematrix (matrix2, matrix1, matrixout)

UNIT II

3D Concepts

→ Depth cueing

Projections

→ Parallel projection

Diagram

Orthographic parallel projection

oblique parallel projection

diagrams

→ Perspective projection

Diagrams

Equations

3D Representation

→ Boundary representations

→ Space-partitioning representation

→ Polygon surfaces

258 PRABHU.S


• Polygon tables

• Geometric tables

� Attribute tables

7. A vertex table,

8. An edge table, and

9. A polygon table.

→ Plane equations

→ Polygon meshes

• Triangle strip

• Quadrilateral mesh

→ Problem

→ Solution

Curved lines and surfaces

→ Quadric surfaces

• Sphere

• Ellipsoid

• Spline

� Spline specifications

Visualization of data sets

→ Visual representations for scalar fields

• diagrams

• Pseudo-color methods

→ Visual representations for vector fields

• Diagrams

→ Visual representations for tensor fields

259 PRABHU.S


• Diagrams

3D Transformation

→ Translation

• Diagram

• Equation

• Matrix form

→ Rotation

• Diagram

• Coordinate-axes rotations

• Equation

• Matrix form

→ Scaling

• Diagram

• Equation

• Matrix form

→ Shear

• Diagram

• Matrix form

→ Reflecetion

• Diagram

• Matrix

260 PRABHU.S


Viewing pipeline

→ Diagram

→ Viewing coordinates

→ Diagram

→ Matrix

Visible surface detection

[ totally 11 methods ]


• Equations

• Diagrams


• Algorithm

25. A-buffer method

UNIT III

Color models

→ Chromaticity diagram

→ Colors representation

→ Diagram

→ Uses of chromaticity diagram

261 PRABHU.S


RBG color model

→ Equation

→ Unit cube diagram

→ Explanation

→ Additive model

YIQ color model

→ Explanation

→ NTSC signals

→ RGB into yIQ

→ YIQ into RGB

CMY color model

→ Video monitors vs printers, plotters:

→ Subtractive process

→ Unit cube diagram

→ Explanation

→ Printing process

→ Conversion of RGB into CMY

→ Conversion of CMY into RGB

HSV color model

→ Explanation

→ Diagrams

262 PRABHU.S


Animation

→ Design of animation sequences





Raster animations

→ Explanation

→ Diagrams

Key-frame systems

→ Morphing

→ Diagrams

OPENGL

→ Advantages:

→ Features:

→ OpenGL operation

→ Diagram

→ Glut

→ Sample program

→ Glut functions

Basic graphics primitives

→ Sample code

→ Format of OpenGL commands

→ OpenGL data types

263 PRABHU.S


→ Sample code

→ Other graphics primitives in OpenGL

→ Example

Drawing 3d scenes with OpenGL

→ Viewing process & graphics pipeline

→ Diagrams

→ Important matrices

vii. Model view matrix

viii. Projection matrix

Diagram

ix. Viewport matrix

Diagram

Drawing three dimensional objects

→ 3d viewing pipeline

→ Diagram

→ OpenGL functions for setting up transformations

→ 3d viewing – model view matrix

→ Sample code

264 PRABHU.S


UNIT IV

Introduction to Shading Model

→ Light sources

→ Black body

→ types of reflection


� Computing the diffuse component

� Diagram

� Lambert’s Law

� diffuse reflection coefficient


Flat Shading

→ OpenGL function

→ Diagram

→ lateral inhibition

Smooth Shading

→ Types

3. Gouraud Shading

i. OpenGL function

ii. Diagrams

4. Phong Shading

i. Diagrams

ii. Drawback:

265 PRABHU.S


Adding texture to faces

→ Diagram

→ Functions

→ Types

• Bitmap textures

Texels


Fakeshape() function

Pasting The Texture On To A Flat Surface

→ Sample code

→ Diagram

→ Mapping a square to rectangle:

→ Diagrams

Adding Shadows Of Objects

→ Diagrams

→ advantages

→ Shadow Buffer

Building a Camera in a Program

→ Camera functions

→ Camera class

→ setModelViewMatrix() function

→ Sliding the Camera

→ Rotating the Camera

266 PRABHU.S


Creating Shaded Objects

→ Diagrams

→ Shading Methods

1. Circulism

2. Blended circulism

3. Dark Blacks

4. Loose Cross Hatching

5. Tight Cross Hatching

6. Powder Shading

Rendering The Texture

→ Diagram

→ Explanation

Drawing Shadows

→ Diagram

→ Explanation

→ Steps

Step 1: Activate and position the shadows

Step 2: Draw the Shadows Only

Step 3: Draw the Shadows from Above

Step 4: Soften the Shadows

Step 5: Create a new Shadow Material

Step 6: Apply the material to a ground polygon

Shadow Mapping:

Advantages:

Disadvantages

267 PRABHU.S


UNIT V

Fractals & self similarity

→ Fractal:

→ Self similarity:

→ Self- similar curves:

→ Exactly self similar:

→ Statistically self similar:

→ Mandelbrot:

→ Koch curve:

o Two generations of the koch curve

o Koch snowflake

Peano curves (or) space-filling curves

→ Diagram

→ Example

Creating an images iterated functions

→ Experimental copier

→ Sierpinski copier

o Diagrams

Mandelbrot sets

→ Iteration theory

→ Mandelbrot sets and iterated function systems

→ Diagram

268 PRABHU.S


Julia sets

→ Diagrams

→ Drawing filled-in julia sets

Random fractals

→ Fractalizing a segment

o Diagram

→ Stages of fractalization

o First stage:

� Diagram

o Second stage:

� Diagram

o Third stage:

� Diagram

→ Calculation of fractalization in a program

• Diagram

• Equation

• Fract() function

• Drawing a fractal curve

• Drawfractal() function

Ray tracing

→ Introduction

→ Diagram

→ Reverse process

→ Features of ray tracing:

269 PRABHU.S


Overview of the ray-tracing process

→ Sample code

→ Computing hit point:

→ Computing color:

o Diagram

→ Object list:

o Diagram

Adding surface texture

→ Solid texture

→ Example:

→ Diagram

Reflections & transparency

→ Diagrams

→ Equation

→ Local component

→ The refraction of light:

o Diagram

o Equation

Boolean operations on objects

→ Csg objects.

→ Boolean operators

• Union

• Intersection

• Difference

270 PRABHU.S


→ Diagram

→ Equations

→ Union of four primitives

→ Diagram

→ Equations

271 PRABHU.S


DIAGRAMS

UNIT I

Stair step Effect (jaggies)

Line

y= m.x + b

Circle

272 PRABHU.S


Ellipse

Line Types

273 PRABHU.S


Pen Brush Options

Fill Styles

Hatch Fill

274 PRABHU.S


Character attribute

UNIT II

Parallel Projection

275 PRABHU.S



276 PRABHU.S


Polygon Table

277 PRABHU.S


Triangle strip

Quadrilateral mesh

Sphere

278 PRABHU.S


ELLIPSOID

SPLINE

Translation

279 PRABHU.S


Rotation

Scaling

280 PRABHU.S


� This sequence of transformations is demonstrated in following fig.

Reflection

281 PRABHU.S


Shear

Viewing Pipeline

Viewing Coordinates

282 PRABHU.S


Transformation from World to Viewing Coordinates

283 PRABHU.S


UNIT III

Chromaticity diagram

RGB color Model

284 PRABHU.S


CMY COLOR MODEL

HSV COLOR MODEL

285 PRABHU.S


Morphing

286 PRABHU.S


Opngl operation:

287 PRABHU.S


Other Graphics Primitives in OPENGL

DRAWING 3D SCENES WITH OPENGL

288 PRABHU.S


Projection matrix

Viewport matrix

289 PRABHU.S


DRAWING THREE DIMENSIONAL OBJECTS

3D Viewing – Model View Matrix

290 PRABHU.S


UNIT IV

Flat Shading

Gourad Shading

291 PRABHU.S


Phong Shading

ADDING TEXTURE TO FACES:

292 PRABHU.S


PROCEDURAL TEXTURE

PASTING THE TEXTURE ON TO A FLAT SURFACE:

MAPPING A SQUARE TO RECTANGLE

293 PRABHU.S


ADDING SHADOWS OF OBJECTS

294 PRABHU.S


Rotating the Camera

CREATING SHADED OBJECTS

295 PRABHU.S


RENDERING THE TEXTURE

PRABHU.S



DRAWING SHADOWS

296

297 PRABHU.S


UNIT V

TWO GENERATIONS OF THE KOCH CURVE

THE FIRST FEW GENERATIONS OF THE KOCH SNOWLAKE

KOCH SNOFLAKE, S3, S4, AND S5

298 PRABHU.S


PEANO CURVES (OR) SPACE-FILLING CURVES

299 PRABHU.S


Low order Hilbert curve

Experimental Copier

300 PRABHU.S


Sierpinski Copier

S-Copier Iterated output

301 PRABHU.S


MANDELBROT SETS

IFS System

302 PRABHU.S


JULIA SETS

Random Fractal

First Stage

303 PRABHU.S


Second Stage

Third Stage

.

Ray Tracing

304 PRABHU.S


Compute color in Ray Tracing

Object list

INTERSTING OF A RAY WITH AN OBJECT

305 PRABHU.S


ADDING SURFACE TEXTURE

Reflection and Transparency

306 PRABHU.S


Contributions of light at each part and its tree structure

The refraction of light:

307 PRABHU.S


Compound Objects built from sphere

Union of 4 primitives

…………………………………………….THE END………………………………………………..

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