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UNIT- I

2D PRIMITIVES

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Line and Curve Drawing

Algorithms

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Line Drawing

y = m . x + b

m = (yendy0) / (xen dx0)

b = y0m . x0x0

y0

xend

yend

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DDA Algorithm

if |m|1

yk+1= yk+ 1

xk+1= xk+ 1/m

x0

y0

xend

yend

x0

y0

xend

yend

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DDA Algorithm

#include

#include

inline int round (const float a) { return int (a + 0.5); }

void lineDDA (int x0, int y0, int xEnd, int yEnd)

{

int dx = xEnd - x0, dy = yEnd - y0, steps, k;

float xIncrement, yIncrement, x = x0, y = y0;

if (fabs (dx) > fabs (dy))

steps = fabs (dx); /* |m|=1 */

xIncrement = float (dx) / float (steps);

yIncrement = float (dy) / float (steps);

setPixel (round (x), round (y));

for (k = 0; k < steps; k++) {

x += xIncrement;

y += yIncrement;

setPixel (round (x), round (y));

}

}

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Bresenhams Line Algorithm

xk

yk

xk+1

yk+1

y

du

dl

xk xk+1

yk

yk+1

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Bresenhams Line Algorithm

#include

#include

/* Bresenham line-drawing procedure for |m| xEnd) {

x = xEnd;

y = yEnd;

xEnd = x0;

}

else {

x = x0;

y = y0;

}

setPixel (x, y);

while (x < xEnd) {

x++;

if (p < 0)

p += twoDy;

else {

y++;

p += twoDyMinusDx;

}

setPixel (x, y);

}

}

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Circle Drawing

Pythagorean Theorem:

x2+ y2= r2

(x-xc)2

+ (y-yc)2

= r2

(xc-r) x (xc+r)

y = ycr2- (x-x

c)2

xc

yc

r(x, y)

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Circle Drawing

change x

change y

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Circle Drawing

using polar coordinates

x = xc+ r . cos

y = yc+ r . sin

change w i th steps ize 1/r

r (x, y)

(xc, yc)

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Circle Drawing

using polar coordinates

x = xc+ r . cos

y = yc+ r . sin

change w i th steps ize 1/r

use symmetry i f >450

r (x, y)

(xc, yc)

(x, y)

(xc, yc)

450

(y, x)(y, -x)

(-x, y)

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Midpoint Circle Algorithm

f(x,y) = x2+ y2- r2

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Midpoint Circle Algorithm#include

class scrPt {public:

GLint x, y;};

void setPixel (GLint x, GLint y){

glBegin (GL_POINTS);glVertex2i (x, y);

glEnd ( );}

void circleMidpoint (scrPt circCtr, GLint radius){

scrPt circPt;

GLint p = 1 - radius;

circPt.x = 0;circPt.y = radius;void circlePlotPoints (scrPt, scrPt);

/* Plot the initial point in each circle quadrant. */circlePlotPoints (circCtr, circPt);

/* Calculate next points and plot in each octant. */while (circPt.x < circPt.y) {

circPt.x++;if (p < 0)p += 2 * circPt.x + 1;

else {circPt.y--;

p += 2 * (circPt.x - circPt.y) + 1;}circlePlotPoints (circCtr, circPt);}

}

void circlePlotPoints (scrPt circCtr, scrPt circPt);{

setPixel (circCtr.x + circPt.x, circCtr.y + circPt.y);setPixel (circCtr.x - circPt.x, circCtr.y + circPt.y);setPixel (circCtr.x + circPt.x, circCtr.y - circPt.y);setPixel (circCtr.x - circPt.x, circCtr.y - circPt.y);

setPixel (circCtr.x + circPt.y, circCtr.y + circPt.x);setPixel (circCtr.x - circPt.y, circCtr.y + circPt.x);setPixel (circCtr.x + circPt.y, circCtr.y - circPt.x);setPixel (circCtr.x - circPt.y, circCtr.y - circPt.x);

}

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OpenGL#include // (or others, depending on the system in use)

void init (void){

glClearColor (1.0, 1.0, 1.0, 0.0); // Set display-window color to white.

glMatrixMode (GL_PROJECTION); // Set projection parameters.gluOrtho2D (0.0, 200.0, 0.0, 150.0);

}

void lineSegment (void)

{ glClear (GL_COLOR_BUFFER_BIT); // Clear display window.

glColor3f (0.0, 0.0, 1.0); // Set line segment color to red.glBegin (GL_LINES);

glVertex2i (180, 15); // Specify line-segment geometry.glVertex2i (10, 145);

glEnd ( );

glFlush ( ); // Process all OpenGL routines as quickly as possible.}

void main (int argc, char** argv){

glutInit (&argc, argv); // Initialize GLUT.glutInitDisplayMode (GLUT_SINGLE | GLUT_RGB); // Set display mode.glutInitWindowPosition (50, 100); // Set top-left display-window position.glutInitWindowSize (400, 300); // Set display-window width and height.glutCreateWindow ("An Example OpenGL Program"); // Create display window.

init ( ); // Execute initialization procedure.glutDisplayFunc (lineSegment); // Send graphics to display window.glutMainLoop ( ); // Display everything and wait.

}

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OpenGL

Point Functions glVertex*( );

* : 2, 3, 4i (integer)

s (short)f (float)d (double)

Ex:glBegin(GL_POINTS);

glVertex2i(50, 100);glEnd();

int p1[ ]={50, 100};glBegin(GL_POINTS);

glVertex2iv(p1);glEnd();

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OpenGL

Line Functions GL_LINES GL_LINE_STRIP

GL_LINE_LOOP

Ex:

glBegin(GL_LINES);glVertex2iv(p1);glVertex2iv(p2);

glEnd();

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OpenGL

glBegin(GL_LINES); GL_LINES GL_LINE_STRIP

glVertex2iv(p1);

glVertex2iv(p2);

glVertex2iv(p3);glVertex2iv(p4);

glVertex2iv(p5);

glEnd();

GL_LINE_LOOP

p1

p1

p1

p2

p3

p2

p2

p4

p3

p3

p5

p4

p4

p5

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Antialiasing

Supersampl ingCount the

number of

subpixels

that overlap

the line path.

Set the intensity

proportional

to this count.

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Antialiasing

Area Sampl ingLine is treated as a rectangle.

Calculate the overlap areasfor pixels.

Set intensity proportional to

the overlap areas.

80% 25%

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Antialiasing

Pixel Sampling

Micropos i t ion ing

Electron beam is shifted 1/2,

1/4, 3/4 of a pixel diameter.

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Line Intensity differences

Change the line drawingalgorithm:

For horizontal andvertical lines use thelowest intensity

For 45olines use thehighest intensity

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2D Transformations

with Matrices

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Matrices

3,32,31,3

3,22,21,2

3,12,11,1

aaa

aaa

aaa

A

A matrix is a rectangular array of numbers.

A general matrix will be represented by an upper-case italicisedletter.

The element on the ith row andjth column is denoted by ai,j. Notethat we start indexing at 1, whereas Cindexes arrays from 0.

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Given two matricesAand Bif we want to add BtoA(that isformA+B) then ifAis (nm), Bmust be (nm), Otherwise,A+Bis not defined.

The addition produces a result, C=A+B, with elements:

Matrices Addition

jijiji BAC ,,,

121086

84736251

8765

4321

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Given two matricesAand Bif we want to multiply BbyA(thatis formAB) then ifAis (nm), Bmust be (mp), i.e., thenumber of columns inAmust be equal to the number of rowsin B. Otherwise,ABis not defined.

The multiplication produces a result, C=AB, with elements:

(Basically we multiply the first row ofAwith the first column ofBand put this in the c1,1element of C.And so on).

Matrices Multiplication

m

k

kjikji baC1

,

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96669555

7644

6233

86

329854

762

Matrices Multiplication (Examples)

26+ 63+ 72=44

6233

86

54

62Undefined!2x2 x 3x2 2!=3

2x2 x 2x4 x 4x4 is allowed. Result is 2x4 matrix

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Unlike scalar multiplication,AB BA

Matrix multiplication distributes over addition:

A(B+C) =AB+AC

Identity matrix for multiplication is defined as I.

The transpose of a matrix,A, is either denotedATorA isobtained by swapping the rows and columns ofA:

Matrices -- Basics

3,23,1

2,22,1

1,21,1

3,22,21,2

3,12,11,1'

aa

aa

aa

Aaaa

aaaA

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2D Geometrical Transformations

Translate

Rotate Scale

Shear

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Translate Points

Recall.. We can translate points in the (x, y) plane to new positionsby adding translation amounts to the coordinates of the points. Foreach point P(x, y) to be moved by dxunits parallel to thexaxis and by dyunits parallel to the yaxis, to the new point P(x, y ). The translation has

the following form:y

x

dyydxx

''

P(x,y)

P(x,y)

dx

dy

In matrix format:

y

x

dd

yx

yx''

If we define the translation matrix , then we have P =P + T.

y

x

d

dT

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Scale Points

Points can be scaled (stretched) by sxalong thexaxis and by syalong the yaxis into the new points by the multiplications:

We can specify how much bigger or smaller by means of a scale factor

To double the size of an object we use a scale factor of 2, to half the size ofan obejct we use a scale factor of 0.5

ysy

xsx

y

x

'

'

P(x,y)

P(x,y)

xsxx

syy

y

y

x

s

s

y

x

y

x

0

0

'

'

If we define , then we have P =SP

y

x

s

sS

0

0

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Rotate Points (cont )

Points can be rotated through an angle about the origin:

cossin

cossinsincos)sin()sin(|'|'

sincos

sinsincoscos

)cos()cos(|'|'

|||'|

yx

lllOPy

yx

ll

lOPx

lOPOP

P(x,y)

P(x,y)

xx

y

y

l

O

y

x

y

x

cossin

sincos

'

'

P =RP

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Review

Translate: P = P+T Scale: P = SP Rotate: P = RP

Spot the odd one out Multiplying versus adding matrix

Ideally, all transformations would be the same.. easier to code

Solution: Homogeneous Coordinates

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Homogeneous Coordinates

For a given 2D coordinates (x, y), we introduce a third dimension:

[x, y, 1]

In general, a homogeneous coordinates for a 2D point has the form:

[x, y, W]

Two homogeneous coordinates [x, y, W] and [x, y, W] are said to be of thesame (or equivalent) if

x= kx eg: [2, 3, 6] = [4, 6, 12]y= ky for some k 0 where k=2W= kW

Therefore any [x, y, W] can be normalised by dividing each element by W:[x/W, y/W, 1]

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Homogeneous Transformations

Now, redefine the translation by using homogeneous coordinates:

Similarly, we have:

y

x

d

d

y

x

y

x

'

'

1100

10

01

1

'

'

y

x

d

d

y

x

y

x

PTP '

1100

00

00

1

'

'

y

x

s

s

y

x

y

x

1100

0cossin

0sincos

1

'

'

y

x

y

x

Scaling Rotation

P = S P P = R P

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Composition of 2D Transformations

1. Additivity of successive translations

We want to translate a point Pto Pby T(dx1, dy1)and then to Pbyanother T(dx2, dy2)

On the other hand, we can define T21= T(dx1, dy1) T(dx2, dy2)first, thenapply T21to P:

where

]),()[,('),('' 112222 PddTddTPddTP yxyxyx

PTP 21''

100

10

01

100

10

01

100

10

01

),(),(

21

21

1

1

2

2

112221

yy

xx

y

x

y

x

yxyx

dd

dd

d

d

d

d

ddTddTT

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T(-1,2) T(1,-1)

(2,1)

(1,3)

(2,2)

100

110001

100

210

101

100

110

101

21T

Examples of Composite 2D Transformations

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Composition of 2D Transformations (cont )

2. Multiplicativity of successive scalings

where

PS

PssSssS

PssSssSP

yxyx

yxyx

21

1122

1122

)],(),([

]),()[,(''

100

0*0

00*

100

00

00

100

00

00

),(),(

12

12

1

1

2

2

112221

yy

xx

y

x

y

x

yxyx

ss

ss

s

s

s

s

ssSssSS

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3. Additivity of successive rotations

where

PR

PRR

PRRP

21

12

12

)]()([

])()[(''

100

0)cos()sin(

0)sin()cos(100

0cossin

0sincos

100

0cossin

0sincos

)()(

1212

1212

11

11

22

22

1221

RRR

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4. Different types of elementary transformations discussed abovecan be concatenated as well.

where

MP

PddTR

PddTRP

yx

yx

)],()([

]),()[('

),()( yx ddTRM

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Consider the following two questions:

1) translate a line segment P1 P2, say, by -1 units in thexdirection

and -2 units in the ydirection.

2). Rotate a line segment P1 P2, say by degrees counter clockwise,about P1.

P1(1,2)

P2(3,3)

P2

P1

P1(1,2)

P2(3,3)P2

P1

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ther Than Point Transformations

Translate Lines: translate both endpoints, then join them.

Scale or Rotate Lines: More complex. For example, consider torotate an arbitrary line about a point P1,

three steps are needed:1). Translate such that P1is at the origin;2). Rotate;3). Translate such that the point at the origin

returns to P1.

T(-1,-2) R()

P1(1,2)

P2(3,3)

P2(2,1)T(1,2)P2

P2

P1

P1P1

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Another Example.

Scale

Translate

Rotate

Translate

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Order Matters

As we said, the order for composition of 2D geometrical transformations matters,because, in general, matrix multiplication is not commutative. However, it is easy to

show that, in the following four cases, commutativity holds:

1). Translation + Translation

2). Scaling + Scaling

3). Rotation + Rotation

4). Scaling (with sx= sy) + Rotation

just to verify case 4:

if sx= sy, M1= M2.

100

0cos*sin*

0sin*cos*

100

0cossin

0sincos

100

00

00

)(),(1

yy

xx

y

x

yx

ss

ss

s

s

RssSM

100

0cos*sin*

0sin*cos*

100

00

00

100

0cossin

0sincos

),()(2

yx

yx

y

x

yx

ss

ss

s

s

ssSRM

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Rigid-Body vs. Affine Transformations

A transformation matrix of the form

where the upper 22 sub-matrix is orthogonal, preserves angles andlengths. Such transforms are called rigid-body transformations,because the body or object being transformed is not distorted in anyway. An arbitrary sequence of rotation and translation matrices

creates a matrix of this form.

The product of an arbitrary sequence of rotation, translations, andscalematrices will cause an affine transformation, which have theproperty of preserving parallelism of lines, but not of lengths and

angles.

100

2221

1211

y

x

trr

trr

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Rigid-Body vs. Affine Transformations (cont )

Shear transformation is also affine.

Unit cube 45 Scale in x , not in y

Rigid- bodyTransformation

AffineTransformation

Shear in the xdirection Shear in the ydirection

100

010

01 a

SHx

100

01

001

bSHy

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2D Output Primitives

Points Lines Circles Ellipses Other curves Filling areas

Text Patterns Polymarkers

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Filling area

Polygons are considered!

1) Scan-Line Filling (between edges)2) Interactive Filling (using an interior

starting point)

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1) Scan-Line Filling (scan

conversion)

Problem: Given the vertices or edges of a

polygon, which are the pixels to beincluded in the area filling?

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Scan-Line filling, contd

Main idea: locate the intersections between the

scan-lines and the edges of the polygon sort the intersection points on each

scan-line on increasing x-coordinates generate frame-buffer positions along

the current scan-line between pairwiseintersection points

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Main idea

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Problemswith intersection points that arevertices:

Basic rule: count them as if each vertex isbeing two points (one to each of the two joiningedges in the vertex)

Exception: if the two edges joining in the

vertex are on opposite sides of the scan-line,then count the vertex only once (require someadditional processing)

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Vertex problem

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Time-consuming to locate the intersection points!

If an edge is crossed by a scan-line, mostprobably also the next scan-line will cross it(the use of coherence properties)

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Each edge is well described by an edge-record:

ymax

x0(initially the x related to ymin) x/y (inverse of the slope) (possibly also x and y)x/y is used for incremental calculations

of the intersection points

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Edge Records

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The intersection point (xn,yn) between an edgeand scan-line ynfollows from the line equationof the edge:

yn= y/x . xn+ b (cp. y = m.x + b)The intersection between the same edge and the

next scan-line yn+1is then given from thefollowing:

yn+1= y/x . xn+1+ band also

yn+1= yn+ 1 = y/x. xn+ b +1

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This gives us:

xn+1= xn+ x/y , n = 0, 1, 2, ..

i.e. the new value of x on the next scan-

line is given by adding the inverse of theslope to the current value of x

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An active list of edge records intersectingwith the current scan-line is sorted on

increasing x-coordinatesThe polygon pixels that are written in theframe buffer are those which arecalculated to be on the current scan-linebetween pairwise x-coordinatesaccording to the active list

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When changing from one scan-line to the next,the active edge list is updated:

a record with ymax< the next scan-line isremoved

in the remaining records, x0is incremented androunded to the nearest integer

an edge with ymin= the next scan-line isincluded in the list

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Scan-Line Filling Example

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2) Interactive Filling

Given the boundaries of a closed surface.By choosing an arbitrary interior point,

the complete interior of the surface willbe filled with the color of the userschoice.

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Interactive Filling, contd

Definition: An area or a boundary is said to be 4-connectedif from an arbitrary point all otherpixels within the area or on the boundary can

be reached by only moving in horizontal orvertical steps.

Furthermore, if it is also allowed to takediagonal steps, the surface or the boundary issaid to be 8-connected.

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4/8-connected

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Interactive Filling, contd

A recursive procedure for filling a 4-connected (8-connected) surface caneasily be defined.

Assume that the surface shall have thesame color as the boundary (can easilybe modified!).

The first interior position (pixel) is choosenby the user.

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Interactive Filling, algorithm

void fill(int x, int y, int fillColor) {int interiorColor;interiorColor = getPixel(x,y);

if (interiorColor != fillColor) {setPixel(x, y, fillColor);fill(x + 1, y, fillColor);

fill(x, y + 1, fillColor);fill(x - 1, y, fillColor);fill(x, y - 1, fillColor); } }

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Inside-Outside test

When is a point an interior point?Odd-Even RuleDraw conceptually a line from a specified point to

a distant point outside the coordinate space;count the number of polygon edges that arecrossed

if odd => interior

if even => exteriorNote! The vertices!

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Text

Representation:* bitmapped (raster)

+ fast- more storage- less good forstyles/sizes

* outlined (lines andcurves)

+ less storage+ good for styles/sizes- slower

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Other output primitives

* pattern (to fill an area)

normally, an n x m rectangularcolor pixel array with a specifiedreference point

* polymarker (marker symbol)

a character representing a point

* (polyline)a connected sequence of line segments

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Attributes

Influence the way a primitive is displayed

Two main ways of introducing attributes:

1) added to primitives parameter liste.g. setPixel(x, y, color)

2) a list of current attributes (to be

updated when changed)e.g setColor(color); setPixel(x, y);

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Attributes for lines

Lines (and curves) are normally infinitely thin type

dashed, dotted, solid, dot-dashed,

pixel mask, e.g. 11100110011 width

problem with the joins; line caps are used to adjustshape

pen/brush shape color (intensity)

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Lines with width

Line caps

Joins

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Attributes for area fill

fill style hollow, solid, pattern, hatch fill,

color pattern

tiling

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Tiling

Tiling = filling surfaces (polygons) with arectangular pattern

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Attributes for characters/strings

style font (typeface) color size (width/height) orientation path

spacing alignment

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Text attributes

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Text attributes, contd

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Color as attribute

Each color has a numerical value, or intensity,based on some color model.

A color model typically consists of three primary

colors, in the case of displays Red, Green andBlue (RGB)For each primary color an intensity can be given,

either 0-255 (integer) or 0-1 (float) yielding the

final color256 different levels of each primary color means3x8=24 bits of information to store

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Color representations

Two different ways of storing a color value:

1) a direct color value storage/pixel

2) indirectly via a color look-up tableindex/pixel (typically 256 or 512different colors in the table)

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Color Look-up Table

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Antialiasing

Aliasing the fact that exact points are

approximated by fixed pixel positions

Antialiasing= a technique thatcompensates for this (more than oneintensity level/pixel is required)

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Antialiasing, a method

A polygon will be studied (as an example).

Area sampling(prefiltering): a pixel that is onlypartly included in the exact polygon, will begiven an intensity that is proportional to theextent of the pixel area that is covered by the

true polygon

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Area sampling

P = polygon intensity

B = background intensity

f = the extent of the pixel

area covered by the truepolygon

pixel intensity =

P*f + B*(1 - f)

Note!Time consuming tocalculate f

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Topics

Clipping

Cohen-Sutherland Line Clipping Algorithm

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Clipping

Why clipping? Not everything defined in the world

coordinates is inside the world window

Where does clipping take place?

OpenGL does it for you BUT, as a CS major, you should know how itis done.

Model ViewportTransformation

Clipping

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Line Clipping

int clipSegment(p1, p2, window)

Input parameters: p1, p2, windowp1, p2: 2D endpoints that define a line

window: aligned rectangle Returned value:

1, if part of the line is inside the window

0, otherwise

Output parameters: p1, p2p1 and/or p2s value might be changed so

that both p1 and p2 are inside the window

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Line Clipping

Example Line RetVal Output

AB

BC

CD

DE

EA

o P1

o P2

o P3

o P4

o P1

o P2o P3

o P4

C h S th l d Li Cli i

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Cohen-Sutherland Line Clipping

Algorithm

Trivial accept and trivial reject If both endpoints within window trivial accept

If both endpoints outside of same boundary ofwindow trivial reject Otherwise

Clip against each edge in turnThrow away clipped off part of line each time

How can we do it efficiently (elegantly)?


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Algorithm

Examples: trivial accept?

trivial reject?

window

L1

L2

L3

L4

L5

L6


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Algorithm

Use region outcode


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Algorithm

outcode[1] (x< Window.left)

outcode[2] (y> Window.top)

outcode[3] (x> Window.right)

outcode[4] (y< Window.bottom)


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Algorithm

Both outcodes are FFFF Trivial accept

Logical AND of two outcodes

FFFF Trivial reject Logical AND of two outcodes = FFFF

Cant tell

Clip against each edge in turnThrow away clipped off part of line each time


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Algorithm

Examples: outcodes?

trivial accept?

trivial reject?

window

L1

L2

L3

L4

L5

L6

Cohen S therland Line Clipping

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Algorithm

int clipSegment(Point2& p1, Point2& p2, RealRect W)do

if(trivial accept) return 1;

else if(trivial reject) return 0;else

if(p1 is inside) swap(p1,p2)if(p1 is to the left) chop against the leftelse if(p1 is to the right) chop against the rightelse if(p1 is below) chop against the bottomelse if(p1 is above) chop against the top

while(1);

Cohen Sutherland Line Clipping

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Algorithm

A segment that requires 4 clips


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Algorithm How do we chop against each boundary?Given P1 (outside) and P2, (A.x,A.y)=?


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Algorithm

Let dx = p1.x - p2.x dy = p1.y - p2.y

A.x = w.r d = p1.y - A.y e = p1.x - w.r

d/dy = e/dxp1.y - A.y = (dy/dx)(p1.x - w.r)

A.y = p1.y - (dy/dx)(p1.x - w.r)

= p1.y + (dy/dx)(w.r - p1.x)As A is the new P1

p1.y += (dy/dx)(w.r - p1.x)p1.x = w.r

Q: Will we havedivided-by-zeroproblem?

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UNIT-IITHREE-

DIMENSIONALCONCEPTS

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3D VIEWING

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3D Viewing-contents

Viewing pipelineViewing coordinatesProjections

View volumes and general projectiontransformations

clipping

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3D Viewing

World coordinate system(where theobjects are modeled and defined)

Viewing coordinate system(viewing objects

with respect to another user defined coordinatesystem) Scene coordinate system(a viewing

coordinate system chosen to be at the centre

of a scene) Object coordinate system(a coordinate

system specific to an object.)

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3D viewing

Simple camera analogy is adopted

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3D viewing-pipeline

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3D viewing

Defining the viewing coordinate system andspecifying the view plane

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3D viewing

First pick up a world coordinate positioncalled the view reference point. This is the origin

of the VC systemPick up the +ve direction for the Zv axis

and the orientation of the view plane byspecifying the view plane normal vector N.

Choose a world coordinate position andthis point establishes the direction for N relativeto either the world or VC origin. The view planenormal vector is the directed line segment.

steps to establish a Viewing coordinate systemor view referencecoordinate system and the view plane

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3D viewing

steps to establish a Viewing coordinate systemor view reference coordinatesystem and the view plane

Some packages allow us to choose a look at

point relative to the view reference point.Or set up a Left handed viewing system and

take the N and the +ve Zv axis from the viewingorigin to the look- at point.

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3D viewing

steps to establish a Viewing coordinate systemor view referencecoordinate system and the view plane

We now choose the view up vector V. It can

be specified as a twist angle about Zv axis.Using N,V U can be specified.

Generally graphics packages allow users tochoose a position of the view plane along the Zv axisby specifying the view plane distance from theviewing origin.

The view plane is always parallel to the XvYvplane.

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3D viewing

To obtain a series of views of a scene we

can keep the view reference point fixed and changethe direction of N or we can fix N direction and movethe view reference point around the scene.

Transformation from world to viewing coordinate system

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3D viewing

(a)

Invert Viewing

z Axis

(b)

Translate Viewing

Origin to World Origin

(c)

Rotate About World x Axis

to Bring Viewing z Axis into

the xz Plane of the World System

(d)

Rotate About the World

y Axis to Align

the Two z Axes

(e)

Rotate About the World

z Axis to Align the Two

Viewing Systems

wx

w

y

wz

wx

wy

wz

wx

wy

wz

wx

wy

wz

wx

wy

wz

vx vy

vz

vz

vyvx

vx

vy

vz v

z

vx

vy

vz

vy

vx

Transformation from world to viewing coordinate system

Mwc,vc=RzRyRx.T

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What Are Projections?

Picture Plane

Objects in

World Space

Our 3-D scenes are all specified in 3-D world coordinatesTo display these we need to generate a 2-D image -projectobjectsonto apicture plane

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Converting From 3-D To 2-D

Projection is just one part of the process of converting from 3-Dworld coordinates to a 2-D image

Clip againstview volume

Project ontoprojection

plane

Transform to2-D devicecoordinates

3-D worldcoordinateoutput

primitives

2-D devicecoordinates

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Types Of Projections

There are two broad classes ofprojection:Parallel: Typically used for architectural and engineering

drawingsPerspective: Realistic looking and used in computer graphics

Perspective ProjectionParallel Projection

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Taxonomy Of Projections

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Types Of Projections

There are two broad classes of projection: Parallel:

preserves relative proportions of objects

accurate views of various sides of an object can be obtaineddoes not give realistic representations of the appearance of a3D objective.

Perspective:produce realistic views but does not preserve relative

proportionsprojections of distant objects are smaller than the projectionsof objects of the same size that are closer to the projectionplane.

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Parallel Projections

Some examples of parallel projections

Orthographic Projection(axonometric)

Orthographic oblique

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Some examples of parallel projections

Isometric projection for a cube

The projectionplane is aligned so that itintersects each coordinateaxes in which the object isdefined (principal axes) atthe same distance from theorigin.

All the principalaxes are foreshortenedequally.

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Transformation equations for an orthographic parallel projections is simple

Any point (x,y,z) in viewing coordinates is transformed to projectioncoordinates as

Xp=X Yp=Y

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Transformation equations for oblique projections is as below.

Oblique projections

11000

0000

0sin10

0cos01

1

1

z

y

x

L

L

w

z

y

x

p

p

p

p

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Transformation equations for oblique projections is as below.

Oblique projections

11000

0000

0sin10

0cos01

1

1

z

y

x

L

L

w

z

y

x

p

p

p

p

An orthographic projection isobtained when L1=0.

In fact the effect of the projectionmatrix is to shear planes ofconstant Z and project them on tothe view plane.

Two common oblique parallel projections:

Cavalierand Cabinet

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Parallel Projections Oblique projections

2 common oblique parallel projections:Cavalier projection

45

1tan

Cabinet projection

4.63

2tan

All lines perpendicular to the projectionplane are projected with no change inlength.

They are more realistic than cavaliarLines perpendicular to the viewingsurface are projected at one-half theirlength.

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Perspective Projections

visual effect is similar to human visual system...has 'perspective foreshortening

size of object varies inversely with distance from the center of projection.

angles only remain intact for faces parallel to projection plane.

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Where u varies from o to 1

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If the view plane is the UV plane itself then Zvp=0.

The projection coordinates become

Xp=X(Zprp/(Zprp-Z))=X(1/(1-Z/Zprp))

Yp=Y(Zprp/(Zprp-Z))=Y(1/(1-Z/Zprp))

If the PRP is selected atthe viewing cooridinateorigin then Zprp=0

The projection coordinates

become

Xp=X(Zvp/Z)

Yp=Y(Zvp/Z)

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There are a number of different kinds of perspective views

The most common are one-point and two point perspectives

One-point perspectiveprojection

Two-point perspectiveprojection

Coordinate description

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Parallel lines that are parallel to the view plane areprojected as parallel lines. The point at which a set of projectedparallel lines appear to converge is called a vanishing point.

If a set of lines are parallel to one of the three principle

axes, the vanishing point is called anprincipal vanishing point.There are at most 3 such points, corresponding to the number of

axes cut by the projection plane.

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View volume

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View volume

Perspective projectionParallelprojection

The size of the view volume depends on the size of the window but theshape depends on the type of projection to be used.

Both near and far planes must be on the same side of the referencepoint.

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View volume

Often the view plane is positioned at the view reference point or on the

front clipping plane while generating parallel projection.Perspective effects depend on the positioning of the projectionreference point relative to the view plane

View volume - PHIGS

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VPN

B

F

Front

Clipping

Plane

View

Plane

Back

Clipping

Plane

Direction of

Propagation

VPN

Front

Clipping

Plane

View

Plane

Back

Clipping

Plane

BF

VRP

VRP

Direction of

Propagation

VPNFront

Clipping

Plane

View

Plane

BackClipping

Plane

BF

VRP

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View volume

In an animation sequence, we canplace the projection reference point at theviewing coordinate origin and put the viewplane in front of the scene.

We set the field of view byadjusting the size of the window relative tothe distance of the view plane from thePRP.

We move through the scene by

moving the viewing reference frame andthe PRP will move with the view referencepoint.

General parallel projection

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N

Direction of

Projection

N

Near

Plane

FarPlane

Window

(a)

Original Orientation

(b)

After Shearing

Direction ofProjection

Window

Near

Plane

Far

Plane

View

Volume

ViewVolume

Paralleltransformation

General parallel projection

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transformation

),0,0(:),,(: ccba

c

bbbcb

caaaca

cc

b

a

b

a

11

11

1

1

0

0

0

0

0

01000

0100

010

001

ShearingLet Vp=(a,b,c) be the projectionvector in viewing coordinates.

The shear transformation can

be expressed asVp=Mparallel.Vp

Where Mparallelis

For an orthographic parallel

projection Mparallel becomes theidentity matrix since a1=b1=0

Parallel

Regularization of Clipping

General perspective projectiontransformation

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Perspective

N N

View

VolumeView

Volume

Far

Near

Window

Far

Near

Window

Center of

Projection

Center of

Projection

(a)

Original Orientation

(b)

After TransformationShearing

g pp g

(View) Volume (Cont)

General perspective projection

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transformation Perspective

Steps

1. Shear the view volume so that thecenterline of the frustum isperpendicular to the view plane

2. Scale the view volume with ascaling factor that depends on 1/z.

A shear operation is to align a generalperspective view volume with theprojection window.

The transformation involves acombination of z-axis shear and atranslation.

Mperspective=Mscale.Mshear

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Clipping

View volume clipping boundaries are planes whoseorientations depend on the type of projection, the projectionwindow and the position of the projection reference point

The process of finding the intersection of a line with oneof the view volume boundaries is simplified if we convert the view

volume before clipping to a rectangular parallelepiped.i.e we first perform the projection transformation which

converts coordinate values in the view volume to orthographicparallel coordinates.

Oblique projection view volumes are converted to arectangular parallelepiped by the shearing operation andperspective view volumes are converted with a combination ofshear and scale transformations.

Clipping-normalized view

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volumes

The normalized view volume is a region defined by the planes

X=0, x=1, y=0, y=1, z=0, z=1


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volumes

There are several advantages to clipping against theunit cube

1. The normalized view volume provides a standardshape for representing any sized view volume.

2. Clipping procedures are simplified and standardizedwith unit clipping planes or the viewport planes.

3. Depth cueing and visible-surface determination aresimplified, since Z-axis always points towards theviewer.

Unit cube

3D viewport

Mapping positions within a rectangularview volume to a three-dimensionalrectangular viewport is accomplishedwith a combination of scaling andtranslation.


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volumes

Unit cube

3D viewport

Mapping positions within arectangular view volume to athree-dimensional rectangularviewport is accomplished with a

combination of scaling andtranslation.

Dx 0 0 Kx

0 Dy 0 Ky

0 0 Dz Kz0 0 0 1

Where

Dx=(xvmax-xvmin)/(xwmax-xwmin) and Kx= xvmin- xwmin Dx

Dy= (yvmax-yvmin)/(ywmax-ywmin) and Ky= yvmin - ywmin Dy

Dz= (zv

max-zv

min)/(zw

max-zw

min) and K

z= zv

min- zw

minD

z

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Viewport clipping

For a line endpoint at position(x,y,z) we assign the bit positionsin the region code from right to leftas

Bit 1 = 1 if x< xvmin (left)Bit 1 = 1 if x< xvmax (right)

Bit 1 = 1 if y< yvmin (below)

Bit 1 = 1 if y< yvmax (above)

Bit 1 = 1 if z< zvmin (front)

Bit 1 = 1 if z< zvmax (back)

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Viewport clipping

For a line segment with endpointsP1(x1,y1,z1) and P2(x2,y2,z2) theparametric equations can be

X=x1+(x2-x1)u

Y=y1+(y2-y1)u

Z=z1+(z2-z1)u

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Hardware implementations

WORLD-COORDINATE

Object descriptions

Transformation Operations

Clipping Operations

Conversion to Device Coordinates

3D Transformations

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2D coordinates 3D coordinates

x

y

x

y

z

xz

y

Right-handed coordinate system:

3D Transformations (cont )

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1. Translation in 3D is a simple extension from that in 2D:

2. Scaling is similarly extended:

1000

100

010

001

),,(z

y

x

zyxd

d

d

dddT

1000

000000

000

),,(z

y

x

zyxs

s

s

sssS

3D Transformations (cont )

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3. The 2D rotation introduced previously is just a 3D rotation aboutthe zaxis.

similarly we have:X

Y

Z

1000

0100

00cossin

00sincos

)(

zR

1000

0cossin00sincos0

0001

)(xR

1000

0cos0sin0010

0sin0cos

)(

yR

Composition of 3D Rotations

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In 3D transformations, the order of a sequence of rotations matters!

1000

0cos0sin

0sinsincoscossin

0sincossincoscos

1000

0cos0sin

0010

0sin0cos

1000

0100

00cossin

00sincos

)()(

yz RR

1000

0cossinsincossin

00cossin

0sincossincoscos

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

)()(

zy RR

)()()()( yzzy RRRR

More Rotations

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We have shown how to rotate about one of the principle axes, i.e. the

axes constituting the coordinate system. There are more we can do,

for example, to perform a rotation about an arbitrary axis:

X

Y

Z

P2(x2, y2 , z2)

P1(x1, y1 , z1)

We want to rotate an object about anaxis in space passing through (x1, y1, z1)

and (x2, y2, z2).

Rotating About An Arbitrary Axis

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Y

Z

P2

P1

1). Translate the object by (-x1, -y1, -z1): T(-x1, -y1, -z1)

X

Y

Z P2

P1

2). Rotate the axis aboutx sothat it lies on thexzplane: Rx()

X

X

Y

Z P2

P1

3). Rotate the axis about y so

that it lies on z: Ry ()

X

Y

ZP2

P1

4). Rotate object about z by : Rz()

Rotating About An Arbitrary Axis (cont )

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After all the efforts, dont forget to undo the rotations and the translation!

Therefore, the mixed matrix that will perform the required task of rotating an

object about an arbitrary axis is given by:

M= T(x1,y1,z1) Rx(-)Ry(-) Rz() Ry() Rx()T(-x1,-y1,-z1)

Finding is trivial, but what about ?

The angle between the zaxis and the

projection of P1 P2on yzplane is .

X

Y

Z

P2

P1

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Composite 3DTransformations

Example of Composite 3D Transformations

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Try to transform the line segments P1P2and P1P3from their start position in (a)

to their ending position in (b).

The first solution is to compose the primitive transformations T, Rx

, Ry

, and Rz

.This approach is easier to illustrate and does offer help on building anunderstanding. The 2nd, more abstract approach is to use the properties ofspecial orthogonal matrices.

y

x

z

y

x

z

P1 P2

P3

P1

P2

P3

(a) (b)


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Breaking a difficult problem into simpler sub-problems:

1.Translate P1to the origin.2. Rotate about the yaxis such that P1P2lies in the (y, z) plane.3. Rotate about thexaxis such that P1P2lies on the zaxis.4. Rotate about the zaxis such that P1P3lies in the (y, z) plane.

y

xz

y

xz

y

xz

P1 P2

P3

P1

P2

P3

y

xz

P1 P2

P3

y

xz

P1P2

P3P3

P2 P1

1

2

34


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1.

2.

1000

100

010

001

),,(

1

1

1

111

z

y

x

zyxT

1000

0sin0cos

0010

0cos0sin

))90((

yR

T

T

T

zzyyxxPzyxTP

zzyyxxPzyxTP

PzyxTP

]1[),,(

]1[),,(

]1000[),,(

1313133111

'

3

1212122111

'

2

1111

'

1

y

xz

P1 P2P3

D1


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3

4.

1000

0cossin0

0sincos0

0001

)(

xRy

xz

P1P2 D2

y

xz

P 1P 2

D3

P 3

)(zR

TRzyxTRRR yxz ),,()90()()( 111Finally, we have the composite matrix:

Vector Rotation

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x

y

x

y

Rotate the vecto r

0

1

u

sincos

01

cossinsincosu

The unit vector along thexaxis is [1, 0]T. After rotating about the origin by, the resulting vector is

Vector Rotation (cont )

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x

y

Rotate the vecto r

1

0

cos

sin

1

0

cossin

sincosv

x

y

v

The above results states that if we try to rotate a vector, originally pointing thedirection of thex (or y)axis, toward a new direction, u (or v), the rotation matrix,R, could be simply written as [u | v] without the need of any explicit knowledgeof , the actual rotation angle.

Similarly, the unit vector along the yaxis is [0, 1]T

. After rotating about theorigin by , the resulting vector is

Vector Rotation (cont )

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The reversed operation of the above rotation is to rotate a vector that is not

originally pointing thex(or y) direction into the direction of the positivexor yaxis. The rotation matrix in this case is R(-), expressed by R-1()

where Tdenotes the transpose.

)(cossin

sincos

)cos()sin(

)sin()cos()(1

T

T

T

Rv

uR

x x

yy

Rotate the vecto ruu

Example

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what is the rotation matrix if one wants the vector Tin the left figure to be

rotated to the direction of u.

T

(2, 3)

TT

u

u

13

3

13

2

32

32

|| 22

13

2

13

313

3

13

2

| vuR

If, on the other hand, one wants the vector uto be rotated to the direction ofthe positivexaxis, the rotation matrix should be

13

2

13

313

3

13

2

T

T

v

uR

Rotation Matrices

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Rotation matrix is or thonormal:

Each row is a unit vector

Each row is perpendicular to the other, i.e. their dot product is zero.

Each vector will be rotated by R() to lie on the positivex and yaxes,respectively. The two column vectors are those into which vectorsalong the positivexand yaxes are rotated.

For orthonormal matrices, we have

1sincos

1)sin(cos

cossin

sincos22

22

R

0)sin(cossincos

)()(1 TRR

Cross Product

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The cross produc torvector product of two vectors, v1and v2, is

another vector:

The cross product of two vectors is orthogonal to both Right-hand rule dictates direction of cross product.

1221

1221

1221

21 )(

yxyx

zxzx

zyzy

vv

v1

v2v1v2

Extension to 3D Cases

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u2

The above examples can be extended to 3D cases.

In 2D, we need to know u, which will be

rotated to the direction of the positivexaxis.uv

x

y

z u1

v=u1u2

In 3D, however, we need to know more than

one vector. See in the left figure, for example,

two vectors, u1and u2are given. If after

rotation, u1is aligned to the positive zaxis, this

will only give us the third column in the rotation

matrix. What about the other two columns?

3D Rotation

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In many cases in 3D, only one vector will be aligned to one of the

coordinate axes, and the others are often not explicitly given. Lets see theexample:y

x

z

y

x

z

P1 P2

P3

P1

P2

P3

Note, in this example, vector P1P2 will be

rotated to the positive zdirection. Hence the

fist columnvector in the rotation matrix is the

normalised P1P2. But what about the other

two columns? After all, P1P3is not perpendi-

cular to P1P2. Well, we can find it by taking

the cross product of P1P2and P1P3. Since

P1P2 P1P3is perpendicular to both P1P2

and P1P3, it will be aligned into the direction

of the positivexaxis.

3D Rotation (cont )y

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And the third direction is decide by the cross

product of the other two directions, which is

P1P2(P1P2P1P2 ).

Therefore, the rotation matrix should be

u

y

xz

P1

P2

P3v

w

21

21

312121

312121

3121

3121

)(

)(

PPPP

PPPPPP

PPPPPP

PPPPPPPP

R

y

x

z

P1P2

P3

u

v

Yaw, Pitch, and Roll

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Imagine three lines running through an airplane and intersecting atright angles at the airplanes centre of gravity.

Roll: rotation around the

front-to-back axis.


side-to-side axis.


vertical axis.

An Example of the Airplane

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Consider the following example. An airplane is oriented such that its nose is

pointing in the positive zdirection, its right wing is pointing in the positivexdirection, its cockpit is pointing in the positive ydirection. We want totransform the airplane so that it heads in the direction given by the vectorDOF(direction of flight), is centre at P,and is not banked.

Solution to the Airplane Example

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First we are to rotate the positive zpdirection into the direction of DOF,

which gives us the third column of the rotation matrix: DOF / |DOF|. Thexpaxis must be transformed into a horizontal vector perpendicular to DOFthat is in the direction of yDOF. The ypdirection is then given byxpzp=DOF (y DOF).

DOF

DOF

DOFyDOF

DOFyDOF

DOFy

DOFyR

)(

)(

Inverses of (2D and) 3D Transformations

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1. Translation:

2. Scaling:

3. Rotation:

4. Shear:

),,(),,(1 zyxzyx dddTdddT

)1

,1

,1

(),,(1

zyx

zyxsss

SsssS

)()()(1 TRRR

),(),(1 yxyx shshSHshshSH

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UNIT-III

GRAPHICS

PROGRAMMING

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Color Models

Color models,contd

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Different meanings of color:

painting

wavelength of visible light

human eye perception

Physical properties of light

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Visible light is part of the electromagnetic radiation (380-750 nm)

1 nm (nanometer) = 10-10m (=10-7cm)

1 (angstrom) = 10 nm

Radiation can be expressed in wavelength () orfrequency (f), c=f, where c=3.1010cm/sec


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White light consists ofa spectrum of allvisible colors


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All kinds of light can bedescribed by theenergy of eachwavelength

The distributionshowing the relationbetween energy andwavelength (or

frequency) is calledenergy spectrum


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This distribution may indicate:

1) a dominant wavelength (or frequency)which is the color of the light (hue), cp.ED

2) brightness (luminance), intensity of thelight (value), cp. the area A

3) purity (saturation), cp. ED- EW


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Energy spectrum for a light source with adominant frequency near the red color

Material properties

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The color of an object depends on the socalled spectral curvesfor transparencyand reflection of the material

The spectral curves describe how light ofdifferent wavelengths are refracted andreflected (cp. the material coefficients

introduced in the illumination models)

Properties of reflected light

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Incident white light upon an object is forsome wavelengths absorbed, for othersreflected

E.g. if all light is absorbed => blackIf all wavelengths but one are absorbed =>

the one color is observed as the color ofthe object by the reflection

Color definitions

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Complementary colorstwo colorscombine to produce white light

Primary colors- (two or) three colors used

for describing other colorsTwo main principles for mixing colors:

additive mixing

subtractive mixing

Additive mixing

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pure colors are put close to each other => a mix on theretina of the human eye (cp. RGB)

overlapping gives yellow, cyan, magenta and white

the typical technique on color displays

Subtractive mixing

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color pigments are mixed directly in someliquid, e.g. ink

each color in the mixture absorbs its specific

part of the incident light the color of the mixture is determined by

subtraction of colored light, e.g. yellow absorbsblue => only red and green, i.e. yellow, will

reach the eye (yellow because of addition)

Subtractive mixing,contd

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primary colors: cyan, magenta andyellow, i.e. CMY

the typical technique in printers/plotters

connection between additive andsubtractive primary colors (cp. the colormodels RGB and CMY)

Additive/subtractive mixing

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Human color seeing

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The retina of the human eye consists of cones(7-8M),tappar, and rods(100-120M), stavar,which are connected with nerve fibres to the

brain

Human color seeing,contd

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Theory: the cones consist of various lightabsorbing material

The light sensitivity of the cones and rods varieswith the wavelength, and between persons

The sum of the energy spectrum of the light the reflection spectrum of the object

the response spectrum of the eyedecides the color perception for a person

Overview of color models

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The human eye can perceive about 382000(!)different colors

Necessary with some kind of classification sys-tem; all using three coordinates as a basis:

1) CIE standard2) RGB color model3) CMY color model (also, CMYK)

4) HSV color model5) HLS color model

CIE standard

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CommissionInternationale deLEclairage (1931)

not a computermodel

each color = aweighted sum of

three imaginaryprimary colors

RGB model

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all colors aregenerated from thethree primaries

various colors areobtained bychanging the amountof each primary

additive mixing(r,g,b), 0r,g,b1

RGB model,contd

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the RGB cube 1 bit/primary => 8 colors, 8 bits/primary => 16M colors

CMY model

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cyan, magenta andyellow are comple-mentary colors ofred,green and blue,

respectively subtractive mixing the typical printer

technique

CMY model,contd

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almost the samecube as with RGB;only black white

the various colorsare obtained byreducing light, e.g. ifred is absorbed =>

green and blue areadded, i.e cyan

RGB vs CMY

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If the intensities are represented as 0r,g,b1

and 0c,m,y1 (also coordinates 0-255 can beused), then the relation between RGB andCMY can be described as:

c

my

1

11

r

gb

CMYK model

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For printing and graphics art industry, CMYis not enough; a fourth primary, K whichstands for black, is added.

Conversions between RGB and CMYK arepossible, although they require someextra processing.

HSV model

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HSV stands for Hue-Saturation-Value

described by a hexcone derived from the RGBcube

HSV model,contd

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Hue (0-360); thecolor, cp. thedominant wave-length (128)

Saturation (0-1); theamount of white(130)

Value (0-1); theamount of black (23)

HSV model,contd

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The numbers given after each primary are

estimates of how many levels a human beingis capable to distinguish between, which (intheory) gives the total number of colornuances:

128*130*23 = 382720

In Computer Graphics, usually enough with:

128*8*15 = 16384

HLS model

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Another model similarto HSV

L stands for Lightness

Color models

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Some more facts about colors:

The distance between two colors in thecolor cube is not a measure of how far

apart the colors are perceptionally!Humans are more sensitive to shifts in

blue (and green?) than, for instance, in

yellow

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COMPUTER

ANIMATIONS

Computer Animations

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Any time sequence of visual changes in a scene. Size, color, transparency, shape, surface texture, rotation,

translation, scaling, lighting effects, morphing, changing cameraparameters(position, orientation, and focal length), particle animation.

Design of animation sequences: Storyboard layout Object definitions Key-frame specifications generation of in-between frames

Computer Animations

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Frame by frame animation Each frame is separately generated.

Object defintion

Objects are defined interms of basic shapes, such as

polygons or splines. In addition the associated movements foreach object are specified along with the shape.

Storyboard

It is an outline of the action

Keyframe Detailed drawing of the scene at a particular instance

Computer Animations

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Inbetweens Intermediate frames (3 to 5 inbetweens for each two key

frames) Motions can be generated using 2D or 3D transformation

Object parameters are stored in database Rendering algorithms are used finally Raster animations: Uses raster operations. Ex: we can animate objects along 2D motion paths using

the color table transformations. Here we predefine the object atsuccessive positions along the motion path, and set thesuccessive blocks of pixel values to color table entries

Computer Animations

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Computer animation languages:A typical task in animation specification is

Scene descriptionincludes position of objects and lightsources, defining the photometric parameters and setting the

camera parameters. Action specificationthis involves layout of motion paths for the

objects and camera. We need viewing and perspectivetransformations, geometric transformations, visible surfacedetection, surface rendering, kinematics etc.,

Keyframe systemsdesigned simply to generate the in-betweens from the user specified key frames.

Computer Animations

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Computer animation languages:A typical task in animation specification is Parameterized systemsallow object motion characteristics to be

specified as part of the object definitions. The adjustableparameters control such object charateristics as degrees offreedom, motion limitations and allowable shape changes.

Scripting systems allow object specifications and animation sequences to be defined with a user-input script. From the script a library of various objects and motions can be constructed.

Computer Animations

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Interpolation techniques Linear

Computer Animations

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Interpolation techniques Non-linear

Computer Animations

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Key frame systems MorphingTransformation of object shapes from one form to another is

called morphing.

Given two keyframes for an object transformation, we first adjust theobject specification in one of the frames so that number of polygon edges

or vertices is the same for the two frames. Let Lk,Lk+1denote the number of line segments in two different frames

K,K+1

Let us define

Lmax

=max(Lk,L

k+1)

Lmin=min(Lk,Lk+1)

Ne= Lmax mod LminNs= int(Lmax / Lmin)

1

2

3

2

1

Key frame K Key frame K+1

Computer Animations

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Steps

1. Dividing Needges of keyframemin into Ns+1 sections

2. Dividing the remaining lines of keyframemin into Nssections

1

2

3

2

1

Key frame K Key frame K+1

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Key frame systems MorphingTransformation of object shapes from one form to another is

called morphing. If we equalize the vertex count, then the similar analysis follows Let Vk,Vk+1denote the number of vertices in two different frames K,K+1 Let us define Vmax=max(Lk,Lk+1) Vmin=min(Lk,Lk+1) Nls= (Vmax1)mod (Vmin1)

Np= int((Vmax1)/ (Vmin1) Steps

1. adding Nppoints to Nlsline sections of keyframemin sections 2. Adding Np-1 points to the remaining edges of keyframemin

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Simulating accelerations Curve fitting techniques are often used to specify the animation paths

between keyframes. To simulate accelerations we can adjust the time spacing for the in-

betweens. For constant speed we use equal interval time spacing for the

inbetweens. suppose we want n in-betweens for keyframes at times t1 and t2.

The time intervals between key frames is then divided into n+1 subintervals, yielding an in-between spacing of

t = t2-t1/(n+1) We can calculate the time for in-betweens as tBj=t1+j t for j=1,2,.,n

t

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Simulating accelerations To model increase or decrease in speeds we use trignometric functions. To model increasing speed, we want the time spacing between frames to

increase so that greater changes in position occur as the object moves faster. We can obtain increase in interval size with the function 1-cos, 0<

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Simulating deccelerations To model increase or decrease in speeds we use trignometric functions. To model decreasing speed, we want the time spacing between

frames to decrease. We can obtain increase in interval size with thefunction

sin, 0<

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Simulating both accelerations and deccelerations To model increase or decrease in speeds we use trignometric functions.

A combination of increasing and decreasing

speeds can be modeled using

(1-cos) 0<

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Motion specifications Direct motion specifications

Here we explicitly give the rotation angles and translation vectors.Then the geometric transformation matrices are applied to transformcoordinate positions.

A bouncing ball can be approximated by a sine curve y(x)=AI(sin(x+0)Ie

-kx

A is the initial amplitude

is the angular frequency

0 is the phase angle K is the damping constant

Computer Animations

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Motion specifications Goal directed systems

We can specify the motions that are to take place in general termsthat abstractly describe the actions, because they determine specificmotion paramters given the goals of the animation.

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Motion specifications Kinematics Kinematic specification

of of a motion can also begiven by simply describingthe motion path which isoften done using splines.

In inverse kinematicswe specify the intital andfinal positions of objects atspecified times and the

motion parameters arecomputed by the system.

Computer Animations

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Motion specifications dynamics

specification of the forces that produce the velocities andaccelerations. Descriptions of object behavior under the influence offorces are generally referred to as a Physically based modeling (.rigidbody systems and non rigid systems such as cloth or plastic)

Ex: magnetic, gravitational, frictional etc

We can also use inverse dynamics to obtain the forces, given the initialand final position of objects and the type of motion.

Computer Animations

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Ideally suited for:Large volumes of objectswind effects, liquid

Cloth animation/draping

Underlying mechanisms are usually:

Particle systems

Mass-spring systems

Physics based animations

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Computer Animations

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Computer Animations

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Some more animationtechniques.

Anticipation and Staging

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Secondary Motion

Computer Animations

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Motion Capture

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Computer Graphicsusing OpenGL

Initial Steps in Drawing Figures

Using Open-GL

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Files: .h, .lib, .dll The entire folder gl is placed in the Include

directory of Visual C++

The individual lib files are placed in the libdirectory of Visual C++ The individual dll files are placed in

C:\Windows\System32

Using Open-GL (2)

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Includes:

(if used)

Include in order given. If you use capital

letters for any file or directory, use themin your include statement also.

Using Open-GL (3)

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Changing project settings: Visual C++6.0 Project menu, Settings entry

In Object/library modules move to the end ofthe line and add glui32.lib glut32.lib glu32.libopengl32.lib (separated by spaces from lastentry and each other)

In Project Options, scroll down to end of boxand add same set of .lib files

Close Project menu and save workspace

Using Open-GL (3)

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Changing Project Settings: Visual C++.NET 2003 Project, Properties, Linker, Command Line

In the white space at the bottom, addglui32.lib glut32.lib glu32.lib opengl32.lib Close Project menu and save your solution

Getting Started Making Pictures

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Graphics display: Entire screen (a);windows system (b); [both have usualscreen coordinates, with y-axis down];

windows system [inverted coordinates](c)

Basic System Drawing Commands

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setPixel(x, y, color) Pixel at location (x, y) gets color specified by

color

Other names: putPixel(), SetPixel(),ordrawPoint() line(x1, y1, x2, y2)

Draws a line between (x1, y1) and (x2, y2) Other names: drawLine()or Line().

Alternative Basic Drawing

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current position (cp), specifies wherethe system is drawing now.

moveTo(x,y)moves the pen invisibly to

the location (x, y) and then updates thecurrent position to this position.

lineTo(x,y)draws a straight line from the

current position to (x, y) and thenupdates the cpto (x, y).

Example: A Square

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moveTo(4, 4); //move to startingcorner

lineTo(-2, 4);

lineTo(-2, -2);

lineTo(4, -2);

lineTo(4, 4);

//close thesquare

Device Independent Graphics and

OpenGL

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Allows same graphics program to be runon many different machine types withnearly identical output.

.dll files must be with program

OpenGL is an API: it controls whateverhardware you are using, and you use itsfunctions instead of controlling the

hardware directly. OpenGL is open source (free).

Event-driven Programs

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Respond to events, such as mouse clickor move, key press, or window reshapeor resize. System manages event queue.

Programmer provides call-backfunctions to handle each event. Call-back functions must be registered

with OpenGL to let it know which

function handles which event. Registering function does *not* call it!

Skeleton Event-driven Program

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// include OpenGL librariesvoid main(){

glutDisplayFunc(myDisplay); // register the redraw functionglutReshapeFunc(myReshape); // register the reshapefunctionglutMouseFunc(myMouse); // register the mouse actionfunctionglutMotionFunc(myMotionFunc); // register the mouse motionfunction

glutKeyboardFunc(myKeyboard); // register the keyboardaction functionperhaps initialize other thingsglutMainLoop(); // enter the unending main loop

}

Callback Functions

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glutDisplayFunc(myDisplay); (Re)draws screen when window opened or another

window moved off it.

glutReshapeFunc(myReshape);

Reports new window width and height for reshapedwindow. (Moving a window does not produce areshape event.)

glutIdleFunc(myIdle);

when nothing else is going on, simply redraws displayusing void myIdle() {glutPostRedisplay();}

Callback Functions (2)

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glutMouseFunc(myMouse); Handles mouse button presses. Knows

mouse location and nature of button (up or

down and which button). glutMotionFunc(myMotionFunc);

Handles case when the mouse is moved withone or more mouse buttons pressed.

Callback Functions (3)

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glutPassiveMotionFunc(myPassiveMotionFunc) Handles case where mouse enters the window with

nobuttons pressed.

glutKeyboardFunc(myKeyboardFunc); Handles key presses and releases. Knows which

key was pressed and mouse location.

glutMainLoop() Runs forever waiting for an event. When one occurs,it is handled by the appropriate callback function.

Libraries to Include

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GL, for which the commands begin with GL; GLUT, the GL Utility Toolkit, opens windows,

develops menus, and manages events. GLU, the GL Utility Library, which provides

high level routines to handle complexmathematical and drawing operations. GLUI, the User Interface Library, which is

completely integrated with the GLUT library. The GLUT functions must be available for GLUI to

operate properly. GLUI provides sophisticated controls and menus toOpenGL applications.

A GL Program to Open a Window

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// appropriate #includes go heresee Appendix 1void main(int argc, char** argv){

glutInit(&argc, argv); // initialize the toolkitglutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);// set the display modeglutInitWindowSize(640,480); // set window sizeglutInitWindowPosition(100, 150);

// set window upper left corner position on screenglutCreateWindow("my first attempt");

// open the screen window (Title: my first attempt)// continued next slide

Part 2 of Window Program

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// register the callback functionsglutDisplayFunc(myDisplay);glutReshapeFunc(myReshape);glutMouseFunc(myMouse);

glutKeyboardFunc(myKeyboard);myInit(); // additional initializations as

necessaryglutMainLoop(); // go into a perpetual

loop} Terminate program by closing window(s) it is using.

What the Code Does

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glutInit (&argc, argv) initializes Open-GLToolkit

glutInitDisplayMode (GLUT_SINGLE |

GLUT_RGB) allocates a single displaybuffer and uses colors to draw

glutInitWindowSize (640, 480) makes the

window 640 pixels wide by 480 pixelshigh

What the Code Does (2)

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glutInitWindowPosition (100, 150) putsupper left window corner at position 100pixels from left edge and 150 pixels

down from top edge glutCreateWindow (my first attempt)

opens and displays the window with the

title my first attempt Remaining functions register callbacks

What the Code Does (3)

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The call-back functions you write areregistered, and then the program entersan endless loop, waiting for events to

occur. When an event occurs, GL calls the

relevant handler function.

Effect of Program

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Drawing Dots in OpenGL

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We start with a coordinate system basedon the window just created: 0 to 679 in xand 0 to 479 in y.

OpenGL drawing is based on vertices

(corners). To draw an object in OpenGL,you pass it a list of vertices. The list starts with glBegin(arg);and ends with

glEnd();

Argdetermines what is drawn. glEnd()sends drawing data down theOpenGL pipeline.

Example

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glBegin (GL_POINTS); glVertex2i (100, 50); glVertex2i (100, 130);

glVertex2i (150, 130);

glEnd(); GL_POINTS is constant built-into Open-

GL (also GL_LINES, GL_POLYGON, ) Complete code to draw the 3 dots is in

Fig. 2.11.

Display for Dots

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What Code Does: GL Function

Construction

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Example of Construction

lV t 2i ( ) t k i t l

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glVertex2i () takes integer values glVertex2d () takes floating point

values

OpenGL has its own data types to makegraphics device-independent

Use these types instead of standard ones

Open-GL Data Types

ffi d t t C/C t O GL t

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suffix data type C/C++ type OpenGL type nameb 8-bit integer signed char GLbyte

s 16-bit integer Short GLshort

i 32-bit integer int or long GLint, GLsizei

f32-bit float Float GLfloat, GLclampf

d 64-bit float Double GLdouble,GLclampd

ub 8-bit unsigned

numberunsigned char GLubyte,GLboolean

us 16-bit unsigned

numberunsigned short GLushort

ui 32-bit unsigned

numberunsigned int or

unsigned longGLuint,Glenum,GLbitfield

Setting Drawing Colors in GL

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glColor3f(red, green, blue);// set drawing color glColor3f(1.0, 0.0, 0.0); // red

glColor3f(0.0, 1.0, 0.0); // green glColor3f(0.0, 0.0, 1.0); // blue glColor3f(0.0, 0.0, 0.0); // black glColor3f(1.0, 1.0, 1.0); // bright white glColor3f(1.0, 1.0, 0.0); // bright yellow glColor3f(1.0, 0.0, 1.0); // magenta

Setting Background Color in GL

C C ( )

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glClearColor (red, green, blue, alpha); Sets background color.Alpha is degree of transparency; use 0.0 for

now. glClear(GL_COLOR_BUFFER_BIT);

clears window to background color

Setting Up a Coordinate System

id I i ( id)

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void myInit(void){

glMatrixMode(GL_PROJECTION);

glLoadIdentity();gluOrtho2D(0, 640.0, 0, 480.0);

}

// sets up coordinate system for windowfrom (0,0) to (679, 479)

Drawing Lines

lB i (GL LINES) //d li

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glBegin (GL_LINES); //draws one line glVertex2i (40, 100); // between 2 vertices glVertex2i (202, 96);

glEnd (); glFlush(); If more than two vertices are specified

between glBegin(GL_LINES) andglEnd() they are taken in pairs, and aseparate line is drawn between eachpair.

Line Attributes

C l thi k ti li

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Color, thickness, stippling. glColor3f()sets color.

glLineWidth(4.0)sets thickness. The default

thickness is 1.0.a). thin lines b). thick lines c). stippledlines

Setting Line Parameters

P l li d P l li t f ti

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Polylines and Polygons: lists of vertices. Polygons are closed (right); polylines

need not be closed (left).

Polyline/Polygon Drawing

lB i (GL LINE STRIP)

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glBegin (GL_LINE_STRIP); // GL_LINE_LOOP to close p

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It Sem7 Cs2401nol