UNIT- I

2D PRIMITIVES

Line and Curve Drawing Algorithms

Line Drawing

y = m . x + b

m = (yend – y0) / (xend – x0)

b = y0 – m . x0x0

y0

xend

yend

DDA Algorithm

if |m|<1

xk+1 = xk + 1

yk+1 = yk + m

if |m|>1

yk+1 = yk + 1

xk+1 = xk + 1/m

x0

y0

xend

yend

x0

y0

xend

yend

DDA Algorithm#include <stdlib.h>#include <math.h>

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 */ else steps = fabs (dy); /* |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)); } }

Bresenham’s Line Algorithm

xk

yk

xk+1

yk+1

ydu

dl

xk xk+1

yk

yk+1

Bresenham’s Line Algorithm#include <stdlib.h>

#include <math.h>

/* Bresenham line-drawing procedure for |m|<1.0 */

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

{

int dx = fabs(xEnd - x0),

dy = fabs(yEnd - y0);

int p = 2 * dy - dx;

int twoDy = 2 * dy,

twoDyMinusDx = 2 * (dy - dx);

int x, y;

/* Determine which endpoint to use as start position. */

if (x0 > 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);

}

}

Circle Drawing

Pythagorean Theorem:

x2 + y2 = r2

(x-xc)2 + (y-yc)2 = r2

(xc-r) ≤ x ≤ (xc+r)

y = yc ± √r2 - (x-xc)2

xc

yc

r(x, y)

Circle Drawing

change x

change y

Circle Drawing using polar coordinates

x = xc + r . cos θ

y = yc + r . sin θ

change θ with step size 1/r

r (x, y)

(xc, yc)

θ

Circle Drawing using polar coordinates

x = xc + r . cos θ

y = yc + r . sin θ

change θ with step size 1/r

use symmetry if θ>450

r (x, y)

(xc, yc)

θ

(x, y)

(xc, yc)

450

(y, x)(y, -x)

(-x, y)

Midpoint Circle Algorithm

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

<0 if (x,y) is inside circle

f(x,y) =0 if (x,y) is on the circle

<0 if (x,y) is outside circle

use symmetry if x>y xk xk+1

yk-1

yk

yk-1/2

Midpoint Circle Algorithm #include <GL/glut.h>

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);}

OpenGL#include <GL/glut.h> // (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.}

OpenGLPoint Functions

• glVertex*( );* : 2, 3, 4 i (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();

OpenGLLine Functions

• GL_LINES

• GL_LINE_STRIP

• GL_LINE_LOOP

Ex: glBegin(GL_LINES); glVertex2iv(p1); glVertex2iv(p2);glEnd();

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

Antialiasing

SupersamplingCount the

number of subpixels that overlap the line path.

Set the intensity proportional to this count.

Antialiasing

Area SamplingLine is treated as a rectangle.

Calculate the overlap areas for pixels.

Set intensity proportional to the overlap areas.

80% 25%

Antialiasing

Pixel Sampling

Micropositioning

Electron beam is shifted 1/2, 1/4, 3/4 of a pixel diameter.

Line Intensity differences

Change the line drawing algorithm:

For horizontal and vertical lines use the lowest intensity

For 45o lines use the highest intensity

2D Transformations with Matrices

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 italicised letter.

The element on the ith row and jth column is denoted by ai,j. Note that we start indexing at 1, whereas C indexes arrays from 0.

Given two matrices A and B if we want to add B to A (that is form A+B) then if A is (nm), B must be (nm), Otherwise, A+B is not defined.

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

Matrices – Addition

jijiji BAC ,,,

1210

86

8473

6251

87

65

43

21

Given two matrices A and B if we want to multiply B by A (that is form AB) then if A is (nm), B must be (mp), i.e., the number of columns in A must be equal to the number of rows in B. Otherwise, AB is not defined.

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

(Basically we multiply the first row of A with the first column of B and put this in the c1,1 element of C. And so on…).

Matrices – Multiplication

m

kkjikji baC

1,

9666

9555

7644

62

33

86

329

854

762

Matrices – Multiplication (Examples)

26+ 63+ 72=44

62

33

86

54

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

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

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 denoted AT or A’ is obtained by swapping the rows and columns of A:

Matrices -- Basics

3,23,1

2,22,1

1,21,1

3,22,21,2

3,12,11,1 '

aa

aa

aa

Aaaa

aaaA

2D Geometrical Transformations

Translate

Rotate Scale

Shear

Translate Points

Recall.. We can translate points in the (x, y) plane to new positions by adding translation amounts to the coordinates of the points. For each point P(x, y) to be moved by dx units parallel to the x axis and by dy

units parallel to the y axis, to the new point P’(x’, y’ ). The translation has the following form:

y

x

dyy

dxx

'

'

P(x,y)

P’(x’,y’)

dx

dy

In matrix format:

y

x

d

d

y

x

y

x

'

'

If we define the translation matrix , then we have P’ =P + T.

y

x

d

dT

Scale Points

Points can be scaled (stretched) by sx along the x axis and by sy along the y axis 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 of an obejct we use a scale factor of 0.5

ysy

xsx

y

x

'

'

P(x,y)

P’(x’,y’)

xsx x

sy y

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

Rotate Points (cont.)

Points can be rotated through an angle about the origin:

cossin

cossinsincos

)sin()sin(|'|'

sincos

sinsincoscos

)cos()cos(|'|'

|||'|

yx

ll

lOPy

yx

ll

lOPx

lOPOP

P(x,y)

P’(x’,y’)

xx’

y’

y

l

O

y

x

y

x

cossin

sincos

'

'

P’ =RP

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

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 the same (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]

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

Composition of 2D Transformations

1. Additivity of successive translations

We want to translate a point P to P’ by T(dx1, dy1) and then to P’’ by another T(dx2, dy2)

On the other hand, we can define T21= T(dx1, dy1) T(dx2, dy2) first, then apply T21 to 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

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

(2,1)

(1,3)

(2,2)

100

110

001

100

210

101

100

110

101

21T

Examples of Composite 2D Transformations

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

Composition of 2D Transformations (cont.)

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

Composition of 2D Transformations (cont.)

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

where

MP

PddTR

PddTRP

yx

yx

)],()([

]),()[('

),()( yx ddTRM

Consider the following two questions:

1) translate a line segment P1 P2, say, by -1 units in the x direction and -2 units in the y direction.

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

P1(1,2)

P2(3,3)

P’2

P’1

P1(1,2)

P2(3,3)P’2

P’1

Other Than Point Transformations…

Translate Lines: translate both endpoints, then join them.

Scale or Rotate Lines: More complex. For example, consider to rotate an arbitrary line about a point P1, three steps are needed:1). Translate such that P1 is 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

Another Example.

Scale

Translate

Rotate

Translate

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 + Translation2). Scaling + Scaling3). Rotation + Rotation4). 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

Rigid-Body vs. Affine Transformations

A transformation matrix of the form

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

The product of an arbitrary sequence of rotation, translations, and scale matrices will cause an affine transformation, which have the property of preserving parallelism of lines, but not of lengths and angles.

1002221

1211

y

x

trr

trr

Rigid-Body vs. Affine Transformations (cont.)

Shear transformation is also affine.

Unit cube 45º Scale in x, not in y

Rigid- bodyTransformation

AffineTransformation

Shear in the x direction Shear in the y direction

100

010

01 a

SH x

100

01

001

bSH y

2D Output Primitives

Points Lines Circles Ellipses Other curves Filling areas Text Patterns Polymarkers

Filling area

Polygons are considered!

1) Scan-Line Filling (between edges)

2) Interactive Filling (using an interior starting point)

1) Scan-Line Filling (scan conversion)

Problem: Given the vertices or edges of a polygon, which are the pixels to be included in the area filling?

Scan-Line filling, cont’d

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 pairwise intersection points

Main idea

Scan-Line filling, cont’d

Problems with intersection points that are vertices:

Basic rule: count them as if each vertex is being two points (one to each of the two joining edges 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 some additional processing)

Vertex problem

Scan-Line filling, cont’d

Time-consuming to locate the intersection points!

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

Scan-Line filling, cont’d

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

Edge Records

Scan-Line filling, cont’d

The intersection point (xn,yn) between an edge and scan-line yn follows from the line equation of the edge:

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

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

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

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

Scan-Line filling, cont’d

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 the slope to the current value of x

Scan-Line filling, cont’d

An active list of edge records intersecting with the current scan-line is sorted on increasing x-coordinates

The polygon pixels that are written in the frame buffer are those which are calculated to be on the current scan-line between pairwise x-coordinates according to the active list

Scan-Line filling, cont’d

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

a record with ymax < ”the next scan-line” is removed

in the remaining records, x0 is incremented and rounded to the nearest integer

an edge with ymin = ”the next scan-line” is included in the list

Scan-Line Filling Example

2) Interactive Filling

Given the boundaries of a closed surface. By choosing an arbitrary interior point, the complete interior of the surface will be filled with the color of the user’s choice.

Interactive Filling, cont’d

Definition: An area or a boundary is said to be 4-connected if from an arbitrary point all other pixels within the area or on the boundary can be reached by only moving in horizontal or vertical steps.

Furthermore, if it is also allowed to take diagonal steps, the surface or the boundary is said to be 8-connected.

4/8-connected

Interactive Filling, cont’d

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

Assume that the surface shall have the same color as the boundary (can easily be modified!).

The first interior position (pixel) is choosen by the user.

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); } }

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 are crossed

if odd => interiorif even => exterior

Note! The vertices!

Text

Representation:* bitmapped (raster) + fast - more storage - less good for

styles/sizes

* outlined (lines and curves)

+ less storage + good for styles/sizes - slower

Other output primitives

* pattern (to fill an area)

normally, an n x m rectangular color pixel array with a specified

reference point

* polymarker (marker symbol)

a character representing a point

* (polyline)

a connected sequence of line segments

Attributes

Influence the way a primitive is displayed

Two main ways of introducing attributes:

1) added to primitive’s 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);

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 adjust shape

pen/brush shape color (intensity)

Lines with width

Line caps

Joins

Attributes for area fill

fill style• hollow, solid, pattern, hatch fill, …

color pattern

• tiling

Tiling

Tiling = filling surfaces (polygons) with a rectangular pattern

Attributes for characters/strings

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

Text attributes

Text attributes, cont’d

Text attributes, cont’d

Text attributes, cont’d

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 and Blue (RGB)

For each primary color an intensity can be given, either 0-255 (integer) or 0-1 (float) yielding the final color

256 different levels of each primary color means 3x8=24 bits of information to store

Color representations

Two different ways of storing a color value:

1) a direct color value storage/pixel

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

Color Look-up Table

Antialiasing

Aliasing ≈ the fact that exact points are approximated by fixed pixel positions

Antialiasing = a technique that compensates for this (more than one intensity level/pixel is required)

Antialiasing, a method

A polygon will be studied (as an example).

Area sampling (prefiltering): a pixel that is only partly included in the exact polygon, will be given an intensity that is proportional to the extent of the pixel area that is covered by the true polygon

Area sampling

P = polygon intensity

B = background intensity

f = the extent of the pixel area covered by the true polygon

pixel intensity =

P*f + B*(1 - f)

Note! Time consuming to calculate f

Topics

Clipping Cohen-Sutherland Line Clipping Algorithm

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 it

is done.

Model

Viewport Transformation

Clipping …

Line Clipping int clipSegment(p1, p2, window)

• Input parameters: p1, p2, window

p1, 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, p2

p1 and/or p2’s value might be changed so that both p1 and p2 are inside the window

Line Clipping

Example• Line RetVal Output

AB

BC

CD

DE

EA

o P1

o P2

o P3

o P4

o P1

o P2

o P3

o P4

Cohen-Sutherland Line Clipping Algorithm

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

• If both endpoints outside of same boundary of window trivial reject

Otherwise• Clip against each edge in turn

Throw away “clipped off” part of line each time

How can we do it efficiently (elegantly)?

Cohen-Sutherland Line Clipping Algorithm

Examples: • trivial accept?

• trivial reject?window

L1

L2L3

L4

L5

L6

Cohen-Sutherland Line Clipping Algorithm

Use “region outcode”

Cohen-Sutherland Line Clipping Algorithm outcode[1] (x < Window.left)

outcode[2] (y > Window.top)

outcode[3] (x > Window.right)

outcode[4] (y < Window.bottom)

Cohen-Sutherland Line Clipping Algorithm

Both outcodes are FFFF• Trivial accept

Logical AND of two outcodes FFFF• Trivial reject

Logical AND of two outcodes = FFFF• Can’t tell

• Clip against each edge in turn

Throw away “clipped off” part of line each time

Cohen-Sutherland Line Clipping Algorithm

Examples: • outcodes?

• trivial accept?

• trivial reject?

window

L1

L2L3

L4

L5

L6

Cohen-Sutherland Line Clipping 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 left else if(p1 is to the right) chop against the right else if(p1 is below) chop against the bottom else if(p1 is above) chop against the top while(1);

Cohen-Sutherland Line Clipping Algorithm

A segment that requires 4 clips

Cohen-Sutherland Line Clipping Algorithm

How do we chop against each boundary?

Given P1 (outside) and P2, (A.x,A.y)=?

Cohen-Sutherland Line Clipping 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/dx

p1.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 have divided-by-zero problem?

UNIT-II

THREE-DIMENSIONAL CONCEPTS

3D VIEWING

3D Viewing-contents•Viewing pipeline

•Viewing coordinates

•Projections

•View volumes and general projection transformations

•clipping

3D Viewing World coordinate system(where the objects are

modeled and defined) Viewing coordinate system(viewing objects with

respect to another user defined coordinate system) 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.)

3D viewing

Simple camera analogy is adopted

3D viewing-pipeline

3D viewing

Defining the viewing coordinate system and specifying the view plane

3D viewing

First pick up a world coordinate position called the view reference point. This is the origin of the VC system

Pick up the +ve direction for the Zv axis and the orientation of the view plane by specifying the view plane normal vector ‘N’.

Choose a world coordinate position and this point establishes the direction for N relative to either the world or VC origin. The view plane normal vector is the directed line segment.

steps to establish a Viewing coordinate system or view reference coordinate system and the view plane

3D viewing

steps to establish a Viewing coordinate system or view reference coordinate system 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 viewing origin to the look- at point.

3D viewing

steps to establish a Viewing coordinate system or view reference coordinate 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 to choose a position of the view plane along the Zv axis by specifying the view plane distance from the viewing origin.

The view plane is always parallel to the XvYv plane.

3D viewing

To obtain a series of views of a scene we can keep the view reference point fixed and change the direction of N or we can fix N direction and move the view reference point around the scene.

3D viewing

(a)Invert Viewing

z Axis

(b)Translate Viewing

Origin to World Origin

(c)Rotate About World x Axisto Bring Viewing z Axis into

the xz Plane of the World System

(d)Rotate About the World

y Axis to Alignthe Two z Axes

(e)Rotate About the Worldz Axis to Align the Two

Viewing Systems

wx

wy

wz

wx

wy

wz

wx

wy

wz

wx

wy

wz

wx

wy

wz

vx vy

vz

vzvyvx

vxvy

vz vz

vxvy

vz

vy

vx

Transformation from world to viewing coordinate system

Mwc,vc=RzRyRx.T

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 - project objects onto a picture plane

Converting From 3-D To 2-D Projection is just one part of the process of converting from 3-D

world coordinates to a 2-D image

Clip against view volume

Project onto projection

plane

Transform to 2-D device coordinates

3-D world coordinate

output primitives

2-D device coordinates

Types Of Projections

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

drawings

Perspective: Realistic looking and used in computer graphics

Perspective ProjectionParallel Projection

Taxonomy Of Projections

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 obtained

does not give realistic representations of the appearance of a 3D objective.

• Perspective:

produce 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.

Parallel Projections Some examples of parallel projections

Orthographic Projection(axonometric)

Orthographic oblique

Parallel Projections Some examples of parallel projections

Isometric projection for a cube

The projection plane is aligned so that it intersects each coordinate axes in which the object is defined (principal axes) at the same distance from the origin.

All the principal axes are foreshortened equally.

Parallel ProjectionsTransformation equations for an orthographic parallel projections is simple

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

Xp=X Yp=Y

Parallel ProjectionsTransformation 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

Parallel ProjectionsTransformation 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 is

obtained when L1=0.

In fact the effect of the projection matrix is to shear planes of constant Z and project them on to the view plane.

•Two common oblique parallel projections:

–Cavalier and Cabinet

Parallel Projections Oblique projections

•2 common oblique parallel projections:Cavalier projection

45

1tan

Cabinet projection

4.63

2tan

All lines perpendicular to the projection plane are projected with no change in length.

They are more realistic than cavaliar

Lines perpendicular to the viewing surface are projected at one-half their length.

Perspective Projectionsvisual 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.

Perspective Projections

Where u varies from o to 1

Perspective Projections

Perspective Projections

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 at the viewing cooridinate origin then Zprp=0

The projection coordinates become

Xp=X(Zvp/Z)

Yp=Y(Zvp/Z)

Perspective Projections There are a number of different kinds of perspective views The most common are one-point and two point perspectives

One-point perspective projection

Two-point perspective projection

Coordinate description

Perspective Projections

Parallel lines that are parallel to the view plane are projected as parallel lines. The point at which a set of projected parallel 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 an principal vanishing point. There are at most 3 such points, corresponding to the number of

axes cut by the projection plane.

View volume

View volume

Perspective projectionParallel projection

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

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

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 projection reference point relative to the view plane

VPN

B

F

FrontClippingPlane

ViewPlane

BackClippingPlane

Direction ofPropagation

VPN

FrontClippingPlane

ViewPlane

BackClippingPlane

BF

VRP

VRP

Direction ofPropagation

VPNFrontClippingPlane

ViewPlane

BackClippingPlane

BF

VRP

View volume - PHIGS

View volume

In an animation sequence, we can place the projection reference point at the viewing coordinate origin and put the view plane in front of the scene.

We set the field of view by adjusting the size of the window relative to the distance of the view plane from the PRP.

We move through the scene by moving the viewing reference frame and the PRP will move with the view reference point.

N

Direction ofProjection

N

NearPlane

FarPlane

Window

(a)Original Orientation

(b)After Shearing

Direction ofProjection

Window

NearPlane

FarPlane

ViewVolume

ViewVolume

ParallelGeneral parallel projection transformation

General parallel projection transformation

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

c

bbbcb

c

aaaca

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 projection vector in viewing coordinates.

The shear transformation can be expressed as

V’p=Mparallel.Vp

Where Mparallel is

For an orthographic parallel projection Mparallel becomes the identity matrix since a1=b1=0

Parallel

Perspective

N N

ViewVolume

ViewVolume

Far

Near

Window

Far

Near

Window

Center ofProjection

Center ofProjection

(a)Original Orientation

(b)After Transformation

Shearing

Regularization of Clipping (View) Volume (Cont’)

General perspective projection transformation

General perspective projection transformation Perspective

Steps

1. Shear the view volume so that the centerline of the frustum is perpendicular to the view plane

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

A shear operation is to align a general perspective view volume with the projection window.

The transformation involves a combination of z-axis shear and a translation.

Mperspective=Mscale.Mshear

Clipping

View volume clipping boundaries are planes whose orientations depend on the type of projection, the projection window and the position of the projection reference point

The process of finding the intersection of a line with one of 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 orthographic parallel coordinates.

Oblique projection view volumes are converted to a rectangular parallelepiped by the shearing operation and perspective view volumes are converted with a combination of shear and scale transformations.

Clipping-normalized view volumes The normalized view volume is a region defined by the planes

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

Clipping-normalized view volumes

There are several advantages to clipping against the unit cube

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

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

3. Depth cueing and visible-surface determination are simplified, since Z-axis always points towards the viewer.

Unit cube

3D viewport

Mapping positions within a rectangular view volume to a three-dimensional rectangular viewport is accomplished with a combination of scaling and translation.

Clipping-normalized view volumes

Unit cube

3D viewport

Mapping positions within a rectangular view volume to a three-dimensional rectangular viewport is accomplished with a combination of scaling and translation.

Dx 0 0 Kx

0 Dy 0 Ky

0 0 Dz Kz

0 0 0 1Where

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

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

Dz= (zvmax-zvmin)/(zwmax-zwmin) and Kz= zvmin- zwmin Dz

Viewport clipping

For a line endpoint at position (x,y,z) we assign the bit positions in the region code from right to left as

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)

Viewport clipping

For a line segment with endpoints P1(x1,y1,z1) and P2(x2,y2,z2) the parametric equations can be

X=x1+(x2-x1)u

Y=y1+(y2-y1)u

Z=z1+(z2-z1)u

Hardware implementations

WORLD-COORDINATE

Object descriptions

Transformation Operations

Clipping Operations

Conversion to Device Coordinates

3D Transformations

2D coordinates 3D coordinates

x

y

x

y

z

xz

y

Right-handed coordinate system:

3D Transformations (cont.)

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

2. Scaling is similarly extended:

1000

100

010

001

),,(z

y

x

zyx d

d

d

dddT

1000

000

000

000

),,(z

y

x

zyx s

s

s

sssS

3D Transformations (cont.)

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

about the z axis.

similarly we have:X

Y

Z

1000

0100

00cossin

00sincos

)(

zR

1000

0cossin0

0sincos0

0001

)(

xR

1000

0cos0sin

0010

0sin0cos

)(

yR

Composition of 3D Rotations

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

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 an

axis in space passing through (x1, y1, z1)

and (x2, y2, z2).

Rotating About An Arbitrary Axis

Y

Z

P2

P1

1). Translate the object by (-x1, -

y1, -z1): T(-x1, -y1, -z1)

X

Y

ZP2

P1

2). Rotate the axis about x so

that it lies on the xz plane: Rx()

X

X

Y

ZP2

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.)

After all the efforts, don’t 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 z axis and the

projection of P1 P2 on yz plane is .

X

Y

Z

P2

P1

Composite 3D Transformations

Example of Composite 3D Transformations

Try to transform the line segments P1P2 and P1P3 from 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 an understanding. The 2nd, more abstract approach is to use the properties of special orthogonal matrices.

y

xz

y

xz

P1P2

P3

P1

P2

P3

(a) (b)

Composition of 3D Transformations

Breaking a difficult problem into simpler sub-problems:1.Translate P1 to the origin.2. Rotate about the y axis such that P1P2 lies in the (y, z) plane.3. Rotate about the x axis such that P1P2 lies on the z axis.4. Rotate about the z axis such that P1P3 lies 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

Composition of 3D Transformations

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 P2

P3

D1

Composition of 3D Transformations

3

4.

1000

0cossin0

0sincos0

0001

)(

xRy

xz

P1P2 D2

y

xz

P1

P2

D3

P3

)(zR

TRzyxTRRR yxz ),,()90()()( 111

Finally, we have the composite matrix:

Vector Rotation

x

y

x

y

Rotate the vector

0

1

u

sin

cos

0

1

cossin

sincosu

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

x

Vector Rotation (cont.)

y

Rotate the vector

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 the direction of the x (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 knowledge of , the actual rotation angle.

Similarly, the unit vector along the y axis is [0, 1]T. After rotating about the origin by , the resulting vector is

Vector Rotation (cont.)

The reversed operation of the above rotation is to rotate a vector that is not originally pointing the x (or y) direction into the direction of the positive x or y axis. The rotation matrix in this case is R(- ), expressed by R-1( )

where T denotes the transpose.

)(cossin

sincos

)cos()sin(

)sin()cos()(1

T

T

T

Rv

uR

x x

yy

Rotate the vector u u

Example

what is the rotation matrix if one wants the vector T in the left figure to be rotated to the direction of u.

T

(2, 3)

u

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 u to be rotated to the direction of the positive x axis, the rotation matrix should be

13

2

13

313

3

13

2

T

T

v

uR

Rotation Matrices

Rotation matrix is orthonormal:• 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 positive x and y axes, respectively. The two column vectors are those into which vectors along the positive x and y axes are rotated.

• For orthonormal matrices, we have

1sincos

1)sin(cos

cossin

sincos22

22

R

0)sin(cossincos

)()(1 TRR

Cross Product

• The cross product or vector product of two vectors, v1 and 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 )(

y x y x

z x z x

z y z y

vv

v1

v2v1 v2

u2

Extension to 3D Cases

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 positive x axis.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, u1 and u2 are given. If after

rotation, u1 is aligned to the positive z axis, this

will only give us the third column in the rotation

matrix. What about the other two columns?

3D Rotation

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. Let’s see the example:

y

xz

y

xz

P1P2

P3

P1

P2

P3

Note, in this example, vector P1P2 will be

rotated to the positive z direction. Hence the

fist column vector in the rotation matrix is the

normalised P1P2. But what about the other

two columns? After all, P1P3 is not perpendi-

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

the cross product of P1P2 and P1P3. Since

P1P2 P1P3 is perpendicular to both P1P2

and P1P3, it will be aligned into the direction

of the positive x axis.

And the third direction is decide by the cross

product of the other two directions, which is

P1P2 (P1P2 P1P2 ).

Therefore, the rotation matrix should be

3D Rotation (cont.)

u

y

xz

P1

P2

P3

v

w

21

21

312121

312121

3121

3121

)(

)(

PP

PPPPPPPP

PPPPPPPPPP

PPPP

R

y

xz

P1P2

P3

u

v

Yaw, Pitch, and Roll

Imagine three lines running through an airplane and intersecting at right angles at the airplane’s centre of gravity.

Roll: rotation around the

front-to-back axis.

Roll: rotation around the

side-to-side axis.

Roll: rotation around the

vertical axis.

An Example of the Airplane

Consider the following example. An airplane is oriented such that its nose is pointing in the positive z direction, its right wing is pointing in the positive x direction, its cockpit is pointing in the positive y direction. We want to transform the airplane so that it heads in the direction given by the vector DOF (direction of flight), is centre at P, and is not banked.

Solution to the Airplane Example

First we are to rotate the positive zp direction into the direction of DOF, which gives us the third column of the rotation matrix: DOF / |DOF|. The xp axis must be transformed into a horizontal vector perpendicular to DOF – that is in the direction of yDOF. The yp direction is then given by xp zp = DOF (y DOF).

DOF

DOF

DOFyDOF

DOFyDOF

DOFy

DOFyR

)(

)(

Inverses of (2D and) 3D Transformations

1. Translation:

2. Scaling:

3. Rotation:

4. Shear:

),,(),,(1zyxzyx dddTdddT

)1

,1

,1

(),,(1

zyxzyx sss

SsssS

)()()(1 TRRR

),(),(1yxyx shshSHshshSH

UNIT-III

GRAPHICS

PROGRAMMING

Color Models

Color models,cont’d

Different meanings of color: painting wavelength of visible light human eye perception

Physical properties of light

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

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

1 Å (angstrom) = 10 nm

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

Physical properties of light

White light consists of a spectrum of all visible colors

Physical properties of light

All kinds of light can be described by the energy of each wavelength

The distribution showing the relation between energy and wavelength (or frequency) is called energy spectrum

Physical properties of light

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 the light (value), cp. the area A

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

Physical properties of light

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

Material properties

The color of an object depends on the so called spectral curves for transparency and reflection of the material

The spectral curves describe how light of different wavelengths are refracted and reflected (cp. the material coefficients introduced in the illumination models)

Properties of reflected light

Incident white light upon an object is for some wavelengths absorbed, for others reflected

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

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

Color definitions

Complementary colors - two colors combine to produce white light

Primary colors - (two or) three colors used for describing other colors

Two main principles for mixing colors: additive mixing subtractive mixing

Additive mixing

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

overlapping gives yellow, cyan, magenta and white the typical technique on color displays

Subtractive mixing

color pigments are mixed directly in some liquid, 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 absorbs blue => only red and green, i.e. yellow, will reach the eye (yellow because of addition)

Subtractive mixing,cont’d

primary colors: cyan, magenta and yellow, i.e. CMY

the typical technique in printers/plotters connection between additive and

subtractive primary colors (cp. the color models RGB and CMY)

Additive/subtractive mixing

Human color seeing

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,cont’d

Theory: the cones consist of various light absorbing material

The light sensitivity of the cones and rods varies with 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

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

Commission Internationale de L’Eclairage (1931)

not a computer model

each color = a weighted sum of three imaginary primary colors

RGB model

all colors are generated from the three primaries

various colors are obtained by changing the amount of each primary

additive mixing (r,g,b), 0≤r,g,b≤1

RGB model,cont’d

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

CMY model

cyan, magenta and yellow are comple-mentary colors of red,green and blue, respectively

subtractive mixing the typical printer

technique

CMY model,cont’d

almost the same cube as with RGB; only black<-> white

the various colors are obtained by reducing light, e.g. if red is absorbed => green and blue are added, i.e cyan

RGB vs CMY

If the intensities are represented as 0≤r,g,b≤1 and 0≤c,m,y≤1 (also coordinates 0-255 can be used), then the relation between RGB and CMY can be described as:

cmy

111

rgb

CMYK model

For printing and graphics art industry, CMY is not enough; a fourth primary, K which stands for black, is added.

Conversions between RGB and CMYK are possible, although they require some extra processing.

HSV model

HSV stands for Hue-Saturation-Value described by a hexcone derived from the RGB

cube

HSV model,cont’d

Hue (0-360°); ”the color”, cp. the dominant wave-length (128)

Saturation (0-1); ”the amount of white” (130)

Value (0-1); ”the amount of black” (23)

HSV model,cont’d

The numbers given after each ”primary” are estimates of how many levels a human being is capable to distinguish between, which (in theory) gives the total number of color nuances:

128*130*23 = 382720

In Computer Graphics, usually enough with:

128*8*15 = 16384

HLS model

Another model similar to HSV

L stands for Lightness

Color models

Some more facts about colors:

The distance between two colors in the color 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

COMPUTER ANIMATIONS

Computer Animations

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

translation, scaling, lighting effects, morphing, changing camera parameters(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

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 for each 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

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 at successive positions along the motion path, and set the successive blocks of pixel values to color table entries

Computer Animations

Computer animation languages:A typical task in animation specification is

Scene description – includes position of objects and light sources, defining the photometric parameters and setting the camera parameters.

Action specification – this involves layout of motion paths for the objects and camera. We need viewing and perspective transformations, geometric transformations, visible surface detection, surface rendering, kinematics etc.,

Keyframe systems – designed simply to generate the in-betweens from the user specified key frames.

Computer Animations Computer animation languages:

A typical task in animation specification is Parameterized systems – allow object motion characteristics to be

specified as part of the object definitions. The adjustable parameters control such object charateristics as degrees of freedom, 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 AnimationsInterpolation techniques Linear

Computer Animations

Interpolation techniques Non-linear

Key frame systems Morphing – Transformation of object shapes from one form to another is called

morphing. Given two keyframes for an object transformation, we first adjust the object

specification in one of the frames so that number of polygon edges or vertices is the same for the two frames.

Let Lk,Lk+1 denote the number of line segments in two different frames K,K+1 Let us define Lmax=max(Lk,Lk+1)

Lmin=min(Lk,Lk+1)

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

Computer Animations

1

2

3’

2’

1’

Key frame K Key frame K+1

Steps 1. Dividing Ne edges of keyframemin into Ns+1 sections

2. Dividing the remaining lines of keyframemin into Nssections

Computer Animations

1

2

3’

2’

1’

Key frame K Key frame K+1

Key frame systems Morphing – Transformation 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+1 denote 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 = (Vmax –1)mod (Vmin –1)

Np = int((Vmax –1) / (Vmin –1) Steps 1. adding Np points to Nls line sections of keyframemin sections 2. Adding Np-1 points to the remaining edges of keyframemin

Computer Animations

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 sub intervals, 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

Computer Animations

t

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< </2 For n-inbetweens the time for the jth inbetween would then be calculated as tBj=t1+t(1-cosj /2(n+1)) j=1,2,…….,n For j=1

tB1=t1+t(1-cos /2(n+1)) For j=1

tB2=t1+t(1-cos 2/2(n+1)) where t is the time difference between any two key frames.

Computer Animations

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 the function

sin, 0< </2 For n-inbetweens the time for the jth inbetween would then be calculated as tBj=t1+t.sinj /2(n+1)) j=1,2,…….,n

Computer Animations

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 ½(1-cos) 0< </2 The time for the jth inbetween is calculated as tBj=t1+t 1-cos j[(n+1)/2) j=1,2,…….,n

Computer Animations

Motion specifications Direct motion specifications Here we explicitly give the rotation angles and translation vectors.

Then the geometric transformation matrices are applied to transform coordinate 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

Motion specifications Goal directed systems We can specify the motions that are to take place in general

terms that abstractly describe the actions, because they determine specific motion paramters given the goals of the animation.

Computer Animations

Motion specifications Kinematics Kinematic specification of of a motion

can also be given by simply describing the motion path which is often done using splines.

In inverse kinematics we specify the intital and final positions of objects at specified times and the motion parameters are computed by the system.

Computer Animations

Motion specifications dynamics specification of the forces that produce the velocities and

accelerations. Descriptions of object behavior under the influence of forces are generally referred to as a Physically based modeling (.rigid body 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 initial

and final position of objects and the type of motion.

Computer Animations

Computer Animations•Ideally suited for:

•Large volumes of objects – wind effects, liquids,

•Cloth animation/draping

•Underlying mechanisms are usually:

•Particle systems

•Mass-spring systems

Physics based animations

Computer Animations

Physics based animations

Computer Animations

Physics based animations

Computer Animations

Some more animation techniques………….

Anticipation and Staging

Computer Animations

Some more animation techniques………….

Secondary Motion

Computer Animations

Some more animation techniques………….

Motion Capture

Computer Graphics using OpenGL

Initial Steps in Drawing Figures

Using Open-GL

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 lib directory of Visual C++

• The individual dll files are placed in C:\Windows\System32

Using Open-GL (2)

Includes:• <windows.h>

• <gl/gl.h>

• <gl/glu.h>

• <gl/glut.h>

• <gl/glui.h> (if used) Include in order given. If you use capital

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

Using Open-GL (3)

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

• In Object/library modules move to the end of the line and add glui32.lib glut32.lib glu32.lib opengl32.lib (separated by spaces from last entry and each other)

• In Project Options, scroll down to end of box and add same set of .lib files

• Close Project menu and save workspace

Using Open-GL (3)

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

• In the white space at the bottom, add glui32.lib glut32.lib glu32.lib opengl32.lib

• Close Project menu and save your solution

Getting Started Making Pictures

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

Basic System Drawing Commands

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

color

• Other names: putPixel(), SetPixel(), or drawPoint()

line(x1, y1, x2, y2) • Draws a line between (x1, y1) and (x2, y2)

• Other names: drawLine() or Line().

Alternative Basic Drawing

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

moveTo(x,y) moves the pen invisibly to the location (x, y) and then updates the current position to this position.

lineTo(x,y) draws a straight line from the current position to (x, y) and then updates the cp to (x, y).

Example: A Square

moveTo(4, 4);//move to starting corner

lineTo(-2, 4); lineTo(-2, -2); lineTo(4, -2); lineTo(4, 4);

//close the square

Device Independent Graphics and OpenGL

Allows same graphics program to be run on many different machine types with nearly identical output.• .dll files must be with program

OpenGL is an API: it controls whatever hardware you are using, and you use its functions instead of controlling the hardware directly.

OpenGL is open source (free).

Event-driven Programs

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

Programmer provides “call-back” functions 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

// include OpenGL librariesvoid main(){

glutDisplayFunc(myDisplay); // register the redraw function glutReshapeFunc(myReshape); // register the reshape function glutMouseFunc(myMouse); // register the mouse action functionglutMotionFunc(myMotionFunc); // register the mouse motion function

glutKeyboardFunc(myKeyboard); // register the keyboard action function…perhaps initialize other things…glutMainLoop(); // enter the unending main loop

}…all of the callback functions are defined here

Callback Functions

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

window moved off it.

glutReshapeFunc(myReshape); • Reports new window width and height for reshaped

window. (Moving a window does not produce a reshape event.)

glutIdleFunc(myIdle); • when nothing else is going on, simply redraws display

using void myIdle() {glutPostRedisplay();}

Callback Functions (2)

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 with

one or more mouse buttons pressed.

Callback Functions (3)

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

no buttons 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 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 complex mathematical 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 to

OpenGL applications.

A GL Program to Open a Window

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

glutInit(&argc, argv); // initialize the toolkit glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);

// set the display mode glutInitWindowSize(640,480); // set window size glutInitWindowPosition(100, 150); // set window upper left corner position on screen

glutCreateWindow("my first attempt"); // open the screen window (Title: my first attempt)

// continued next slide

Part 2 of Window Program

// register the callback functions glutDisplayFunc(myDisplay); glutReshapeFunc(myReshape); glutMouseFunc(myMouse); glutKeyboardFunc(myKeyboard);

myInit(); // additional initializations as necessary

glutMainLoop(); // go into a perpetual loop} Terminate program by closing window(s) it is using.

What the Code Does

glutInit (&argc, argv) initializes Open-GL Toolkit

glutInitDisplayMode (GLUT_SINGLE | GLUT_RGB) allocates a single display buffer and uses colors to draw

glutInitWindowSize (640, 480) makes the window 640 pixels wide by 480 pixels high

What the Code Does (2)

glutInitWindowPosition (100, 150) puts upper left window corner at position 100 pixels 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)

The call-back functions you write are registered, and then the program enters an endless loop, waiting for events to occur.

When an event occurs, GL calls the relevant handler function.

Effect of Program

Drawing Dots in OpenGL We start with a coordinate system based

on the window just created: 0 to 679 in x and 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();• Arg determines what is drawn.• glEnd() sends drawing data down the OpenGL

pipeline.

Example

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

What Code Does: GL Function Construction

Example of Construction

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

values

OpenGL has its own data types to make graphics device-independent• Use these types instead of standard ones

Open-GL Data Typessuffix data type C/C++ type OpenGL type name

b 8-bit integer signed char GLbyte

s 16-bit integer Short GLshort

i 32-bit integer int or long GLint, GLsizei

f 32-bit float Float GLfloat, GLclampf

d 64-bit float Double GLdouble,GLclampd

ub 8-bit unsigned number

unsigned char GLubyte,GLboolean

us 16-bit unsigned number

unsigned short GLushort

ui 32-bit unsigned number

unsigned int or unsigned long

GLuint,Glenum,GLbitfield

Setting Drawing Colors in GL

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

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

void myInit(void){

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

}// sets up coordinate system for window

from (0,0) to (679, 479)

Drawing Lines

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) and glEnd() they are taken in pairs, and a separate line is drawn between each pair.

Line Attributes

Color, thickness, stippling. glColor3f() sets color. glLineWidth(4.0) sets thickness. The default

thickness is 1.0.

a). thin lines b). thick lines c). stippled lines

Setting Line Parameters

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

need not be closed (left).

Polyline/Polygon Drawing

glBegin (GL_LINE_STRIP); // GL_LINE_LOOP to close polyline (make

it a polygon)• // glVertex2i () calls go here

glEnd (); glFlush (); A GL_LINE_LOOP cannot be filled with

color

Examples

Drawing line graphs: connect each pair of (x, f(x)) values

Must scale and shift

Examples (2)

Drawing polyline from vertices in a file• # polylines

• # vertices in first polyline

• Coordinates of vertices, x y, one pair per line

• Repeat last 2 lines as necessary

File for dinosaur available from Web site Code to draw polylines/polygons in Fig.

2.24.

Examples (3)

Examples (4)

Parameterizing Drawings: allows making them different sizes and aspect ratios

Code for a parameterized house is in Fig. 2.27.

Examples (5)

Examples (6)

Polyline Drawing Code to set up an array of vertices is in

Fig. 2.29. Code to draw the polyline is in Fig. 2.30.

Relative Line Drawing

Requires keeping track of current position on screen (CP).

moveTo(x, y); set CP to (x, y) lineTo(x, y);draw a line from CP to (x, y), and

then update CP to (x, y). Code is in Fig. 2.31. Caution! CP is a global variable, and therefore

vulnerable to tampering from instructions at other points in your program.

Drawing Aligned Rectangles

glRecti (GLint x1, GLint y1, GLint x2, GLint y2); // opposite corners; filled with current color; later rectangles are drawn on top of previous ones

Aspect Ratio of Aligned Rectangles

Aspect ratio = width/height

Filling Polygons with Color

Polygons must be convex: any line from one boundary to another lies inside the polygon; below, only D, E, F are convex

Filling Polygons with Color (2)

glBegin (GL_POLYGON);• //glVertex2f (…); calls go here

glEnd (); Polygon is filled with the current drawing

color

Other Graphics Primitives

GL_TRIANGLES, GL_TRIANGLE_STRIP, GL_TRIANGLE_FAN

GL_QUADS, GL_QUAD_STRIP

Simple User Interaction with Mouse and Keyboard

Register functions:• glutMouseFunc (myMouse);

• glutKeyboardFunc (myKeyboard); Write the function(s) NOTE that any drawing you do when you

use these functions must be done IN the mouse or keyboard function (or in a function called from within mouse or keyboard callback functions).

Example Mouse Function

void myMouse(int button, int state, int x, int y);

Button is one of GLUT_LEFT_BUTTON, GLUT_MIDDLE_BUTTON, or GLUT_RIGHT_BUTTON.

State is GLUT_UP or GLUT_DOWN. X and y are mouse position at the time of

the event.

Example Mouse Function (2)

The x value is the number of pixels from the left of the window.

The y value is the number of pixels down from the top of the window.

In order to see the effects of some activity of the mouse or keyboard, the mouse or keyboard handler must call either myDisplay() or glutPostRedisplay().

Code for an example myMouse() is in Fig. 2.40.

Polyline Control with Mouse

Example use:

Code for Mouse-controlled Polyline

Using Mouse Motion Functions

glutMotionFunc(myMovedMouse); // moved with button held down

glutPassiveMotionFunc(myMovedMouse); // moved with buttons up

myMovedMouse(int x, int y); x and y are the position of the mouse when the event occurred.

Code for drawing rubber rectangles using these functions is in Fig. 2.41.

Example Keyboard Function

void myKeyboard(unsigned char theKey, int mouseX, int mouseY){

GLint x = mouseX; GLint y = screenHeight - mouseY; // flip y value switch(theKey) {case ‘p’: drawDot(x, y); break;// draw dot at mouse position case ‘E’: exit(-1); //terminate the program

default: break; // do nothing}

}

Example Keyboard Function (2) Parameters to the function will always be

(unsigned char key, int mouseX, int mouseY). The y coordinate needs to be flipped by

subtracting it from screenHeight. Body is a switch with cases to handle active

keys (key value is ASCII code). Remember to end each case with a break!

Using Menus

Both GLUT and GLUI make menus available.

GLUT menus are simple, and GLUI menus are more powerful.

We will build a single menu that will allow the user to change the color of a triangle, which is undulating back and forth as the application proceeds.

GLUT Menu Callback Function

Int glutCreateMenu(myMenu); returns menu ID void myMenu(int num); //handles choice num void glutAddMenuEntry(char* name, int value); //

value used in myMenu switch to handle choice void glutAttachMenu(int button); // one of

GLUT_RIGHT_BUTTON, GLUT_MIDDLE_BUTTON, or GLUT_LEFT_BUTTON • Usually GLUT_RIGHT_BUTTON

GLUT subMenus

Create a subMenu first, using menu commands, then add it to main menu.• A submenu pops up when a main menu item is selected.

glutAddSubMenu (char* name, int menuID); // menuID is the value returned by glutCreateMenu when the submenu was created

Complete code for a GLUT Menu application is in Fig. 2.44. (No submenus are used.)

GLUI Interfaces and Menus

GLUI Interfaces

An example program illustrating how to use GLUI interface options is available on book web site.

Most of the work has been done for you; you may cut and paste from the example programs in the GLUI distribution.

UNIT-IV

RENDERING

Polygon shading model

Flat shading - compute lighting once and assign the color to the whole (mesh) polygon

Flat shading

Only use one vertex normaland material property to compute the color for the polygon

Benefit: fast to compute Used when:

• Polygon is small enough

• Light source is far away (why?)

• Eye is very far away (why?)

Mach Band Effect

Flat shading suffers from “mach band effect” Mach band effect – human eyes accentuate the

discontinuity at the boundary

Side view of a polygonal surface

perceived intensity

Smooth shading

Fix the mach band effect – remove edge discontinuity

Compute lighting for more points on each face

Flat shading Smooth shading

Smooth shading

Two popular methods: • Gouraud shading (used by OpenGL)

• Phong shading (better specular highlight, not in OpenGL)

Gouraud Shading

The smooth shading algorithm used in OpenGL

glShadeModel(GL_SMOOTH) Lighting is calculated for each of the polygon vertices Colors are interpolated for interior pixels

Gouraud Shading

Per-vertex lighting calculation Normal is needed for each vertex Per-vertex normal can be computed by

averaging the adjust face normalsn

n1 n2

n3 n4n = (n1 + n2 + n3 + n4) / 4.0

Gouraud Shading

Compute vertex illumination (color) before the projection transformation

Shade interior pixels: color interpolation (normals are not needed)

C1

C2 C3

Ca = lerp(C1, C2) Cb = lerp(C1, C3)

Lerp(Ca, Cb)

for all scanlines

* lerp: linear interpolation

Gouraud Shading

Linear interpolation

Interpolate triangle color: use y distance to interpolate the two end points in the scanline, and use x distance to interpolate interior

pixel colors

a b

v1 v2x

x = a / (a+b) * v1 + b/(a+b) * v2

Gouraud Shading Problem

Lighting in the polygon interior can be inaccurate

Phong Shading

Instead of interpolation, we calculate lighting for each pixel inside the polygon (per pixel lighting)

Need normals for all the pixels – not provided by user Phong shading algorithm interpolates the normals

and compute lighting during rasterization (need to map the normal back to world or eye space though)

Phong Shading

Normal interpolation

• Slow – not supported by OpenGL and most graphics hardware

n1

n2

n3

nb = lerp(n1, n3)na = lerp(n1, n2)

lerp(na, nb)

UNIT-V

FRACTALS

Fractals

Fractals are geometric objects. Many real-world objects like ferns are

shaped like fractals. Fractals are formed by iterations. Fractals are self-similar. In computer graphics, we use fractal

functions to create complex objects.

Koch Fractals (Snowflakes)

Iteration 0 Iteration 1 Iteration 2 Iteration 3

Generator1/3 1/3

1/3 1/3

1

Fractal Tree

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

Generator

Fractal Fern

Generator

Iteration 0 Iteration 1 Iteration 2 Iteration 3

Add Some Randomness

The fractals we’ve produced so far seem to be very regular and “artificial”.

To create some realism and variability, simply change the angles slightly sometimes based on a random number generator.

For example, you can curve some of the ferns to one side.

For example, you can also vary the lengths of the branches and the branching factor.

Terrain (Random Mid-point Displacement)

Given the heights of two end-points, generate a height at the mid-point.

Suppose that the two end-points are a and b. Suppose the height is in the y direction, such that the height at a is y(a), and the height at b is y(b).

Then, the height at the mid-point will be: ymid = (y(a)+y(b))/2 + r, where

• r is the random offset This is how to generate the random offset r:

r = srg|b-a|, where • s is a user-selected “roughness” factor, and• rg is a Gaussian random variable with mean 0 and variance 1

How to generate a random number with Gaussian (or normal) probability distribution

// given random numbers x1 and x2 with equal distribution from -1 to 1// generate numbers y1 and y2 with normal distribution centered at 0.0// and with standard deviation 1.0.void Gaussian(float &y1, float &y2) {

float x1, x2, w;

do {x1 = 2.0 * 0.001*(float)(rand()%1000) - 1.0;x2 = 2.0 * 0.001*(float)(rand()%1000) - 1.0;w = x1 * x1 + x2 * x2;

} while ( w >= 1.0 );

w = sqrt( (-2.0 * log( w ) ) / w );y1 = x1 * w;y2 = x2 * w;

}

Procedural Terrain Example

Building a more realistic terrain

Notice that in the real world, valleys and mountains have different shapes.

If we have the same terrain-generation algorithm for both mountains and valleys, it will result in unrealistic, alien-looking landscapes.

Therefore, use different parameters for valleys and mountains.

Also, can manually create ridges, cliffs, and other geographical features, and then use fractals to create detail roughness.

Fractals

Infinite detail at every point Self similarity between parts and overall features of the object Zoom into Euclidian shape

• Zoomed shape see more detail

• eventually smooths Zoom in on fractal

• See more detail

• Does not smooth Model

• Terrain, clouds water, trees, plants, feathers, fur, patterns General equation P1=F(P0), P2 = F(P1), P3=F(P2)…

• P3=F(F(F(P0)))

Self similar fractals

Parts are scaled down versions of the entire object• use same scaling on subparts

• use different scaling factors for subparts

Statistically self-similar• Apply random variation to subparts

• Trees, shrubs, other vegetation

Fractal types

Statistically self-affine • random variations

• Sx<>Sy<>Sz• terrain, water, clouds

Invariant fractal sets• Nonlinear transformations

• Self squaring fractals• Julia-Fatou set

• Squaring function in complex space

• Mandelbrot set

• Squaring function in complex space

• Self-inverse fractals• Inversion procedures

Julia-Fatou and Mandelbrot

x=>x2+c

• x=a+bi

• Complex number Modulus

• Sqrt(a2+b2)

• If modulus < 1

• Squaring makes it go toward 0

• If modulus > 1

• Squaring falls towards infinity

If modulus=1

• Some fall to zero

• Some fall to infinity

• Some do neither

• Boundary between numbers which fall to zero and those which fall to infinity

• Julia-Fatou Set

Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition

Julia-Fatou

Julia Fatou and Mandelbrot con’d

Shape of the Julia-Fatou set based on c To get Mandelbrot set – set of non-diverging points

• Correct method

• Compute the Julia sets for all possible c

• Color the points black when the set is connected and white when it is not connected

• Approximate method

• Foreach value of c, start with complex number 0=0+0i

• Apply to x=>x2+c

• Process a finite number of times (say 1000)

• If after the iterations is is outside a disk defined by modulus>100, color the points of c white, otherwise color it black.

Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition

Constructing a deterministic self-similar fractal

Initiator• Given geometric shape

Generator• Pattern which replaces subparts of initiator

Koch Curve

Initiator generator First iteration

Fractal dimension

D=fractal dimension

• Amount of variation in the structure

• Measure of roughness or fragmentation of the object

• Small d-less jagged

• Large d-more jagged Self similar objects

• nsd=1 (Some books write this as ns-d=1)

• s=scaling factor

• n number of subparts in subdivision

• d=ln(n)/ln(1/s)

• [d=ln(n)/ln(s) however s is the number of segments versus how much the main segment was reduced

• I.e. line divided into 3 segments. Instead of saying the line is 1/3, say instead there are 3 sements. Notice that 1/(1/3) = 3]

• If there are different scaling factors

•

• Skd=1

K=1

n

Figuring out scaling factorsI prefer: ns-d=1 :d=ln(n)/ln(s)

Dimension is a ratio of the (new size)/(old size)• Divide line into n identical

segments• n=s

• Divide lines on square into small squares by dividing each line into n identical segments• n=s2 small squares

• Divide cube• Get n=s3 small cubes

Koch’s snowflake• After division have 4 segments

• n=4 (new segments)

• s=3 (old segments)

• Fractal Dimension

• D=ln4/ln3 = 1.262

• For your reference: Book method

• n=4

• Number of new segments

• s=1/3

• segments reduced by 1/3

• d=ln4/ln(1/(1/3))

Sierpinski gasket Fractal Dimension

Divide each side by 2• Makes 4 triangles

• We keep 3

• Therefore n=3• Get 3 new triangles from 1 old

triangle

• s=2 (2 new segments from one old segment)

Fractal dimension • D=ln(3)/ln(2) = 1.585

Cube Fractal Dimension

Apply fractal algorithm• Divide each side by 3

• Now push out the middle face of each cube

• Now push out the center of the cube What is the fractal dimension?

• Well we have 20 cubes, where we used to have 1• n=20

• We have divided each side by 3• s=3

• Fractal dimension ln(20)/ln(3) = 2.727

Image from Angel book

Language Based Models of generating images

Typical Alphabet {A,B,[,]} Rules

• A=> AA

• B=> A[B]AA[B] Starting Basis=B Generate words

• Represents sequence of segments in graph structure

• Branch with brackets

• Interesting, but I want a tree

B A[B]AA[B] AA[A[B]AA[B]]AAAA[A[B]AA[B]]

A

AA

B

B

A

AAB

AA

B

AAAA

A

B

AA

B

Language Based Models of generating images con’d

Modify Alphabet {A,B,[,],(,)}

Rules• A=> AA

• B=> A[B]AA(B)

• [] = left branch () = right branchStarting Basis=B

Generate words• Represents sequence of

segments in graph structure

• Branch with brackets

B A[B]AA(B) AA[A[B]AA(B)]AAAA(A[B]AA(B))

A

AA

B

B

A

AAB

AA

B

AAAA

AB

AA

B

Language Based models have no inherent geometry

Grammar based model requires

• Grammar

• Geometric interpretation Generating an object from the word is a

separate process

• examples

• Branches on the tree drawn at upward angles

• Choose to draw segments of tree as successively smaller lengths

• The more it branches, the smaller the last branch is

• Draw flowers or leaves at terminal nodes

A

AAB

AA

B

AAAA

AB

AA

B

Grammar and Geometry

Change branch size according to depth of graph

Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition

Particle Systems

System is defined by a collection of particles that evolve over time

• Particles have fluid-like properties

• Flowing, billowing, spattering, expanding, imploding, exploding

• Basic particle can be any shape

• Sphere, box, ellipsoid, etc

• Apply probabilistic rules to particles

• generate new particles

• Change attributes according to age

• What color is particle when detected?

• What shape is particle when detected?

• Transparancy over time?

• Particles die (disappear from system)

• Movement

• Deterministic or stochastic laws of motion

• Kinematically

• forces such as gravity

Particle Systems modeling

Model

• Fire, fog, smoke, fireworks, trees, grass, waterfall, water spray. Grass

• Model clumps by setting up trajectory paths for particles Waterfall

• Particles fall from fixed elevation

• Deflected by obstacle as splash to ground

• Eg. drop, hit rock, finish in pool

• Drop, go to bottom of pool, float back up.

Physically based modeling

Non-rigid object

• Rope, cloth, soft rubber ball, jello Describe behavior in terms of external and internal forces

• Approximate the object with network of point nodes connected by flexible connection

• Example springs with spring constant k

• Homogeneous object

• All k’s equal Hooke’s Law

• Fs=-k x

• x=displacement, Fs = restoring force on spring Could also model with putty (doesn’t spring back) Could model with elastic material

• Minimize strain energy

kk

kk

“Turtle Graphics” Turtle can

• F=Move forward a unit

• L=Turn left

• R=Turn right Stipulate turtle directions, and

angle of turns Equilateral triangle

• Eg. angle =120

• FRFRFR What if change angle to 60

degrees• F=> FLFRRFLF

• Basis F• Koch Curve (snowflake)

• Example taken from Angel book

Using turtle graphics for trees

Use push and pop for side branches []

F=> F[RF]F[LF]F Angle =27 Note spaces ONLY for

readability F[RF]F[LF]F [RF[RF]F[LF]F]

F[RF]F[LF]F [LF[RF]F[LF]F] F[RF]F[LF]F