Computer Graphics

CS 385

January 31, 2005

Fractals

Some definitions

• Object which is self-similar at all scales. Regardless of scale the same level of detail and appearance is present

• A System having similar detail at all scales, leading to intricate patterns and unexpected features. A system with a non-integer dimension.

• Computer-generated images corresponding to mathematical equations, that repeat self-similar patterns at infinitely receding levels of organization.

The Mandelbrot Set

A Julia Set

The Sierpinski Gasket

This object is defined recursively but its definition also has a random component.

Nevertheless, the process always converges to a geometric object that looks the same.

The theme here is: Order in Chaos

Non-Formal Description

1. Draw a triangle

2. Pick a point p in the triangle at random and draw it

3. Pick a vertex v of the triangle at random

4. Find the point q halfway between v and p

5. Go back to step 2 with q playing the role of p

Representing Points in OpenGL

In OpenGL, a point is often called a vertex

There is a family of functions that start out

glVertex and can be used for various data types in either 2 or 3 dimensions.

Representing Points in OpenGL

Examples of glVertex are glVertex2f and glVertex3i

If the datatype is represented by the letter v, then this indicates that the function takes a pointer to an array as its argument.

#include <GL/glut.h>

void init(void){

glClearColor(1.0,1.0,1.0,1.0);glColor3f(0.0,0.0,0.0);

}

void display(void){ glClear(GL_COLOR_BUFFER_BIT); glBegin(GL_POINTS); glVertex2f(0,0); glEnd(); glFlush();}

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

glutInit(&argc, argv);glutCreateWindow(“A Point”);glutDisplayFunc(display);init();glutMainLoop();return 0;

}

More Examples

glBegin(GL_POINTS);

glVertex2f(0,0);

glEnd();

glBegin(GL_LINES);

glVertex2f(0,0);

glVertex2f(1,1);

glEnd();

glBegin(GL_LINES); glVertex2f(1,1); glVertex2f(-1,-1); glVertex2f(-1,1); glVertex2f(1,-1);glEnd();

Back to the Sierpinski Gasket

Whenever the window needs to be redrawn, the OpenGL engine calls a function specified by the function

glutDisplayFunc

This function takes a function which take a void argument and returns a void. By convention, the function is often called display. Lets look at an implementation of a display function for the Sierpinski Gasket

typedef GLfloat point2[2];void display(void){

//Create an arbitrary trianglepoint2 vertices[3]={ {0.0, 0.0}, {250.0,500.0},{500.0,0.0}};

//Pick an arbitray point in the trianglestatic point2 p = {75.0, 50.0};

int j; for (int k=0; k<5000; k++)

{j=rand()%3; //pick a random vertex from 0,1 and 2//Compute the new pointp[0] = (p[0] + vertices[j][0])/2;p[1] = (p[1] + vertices[j][1])/2;

//Display the new pointglBegin(GL_POINTS); glVertex2fv(p);

glEnd();}glFlush();

}

glBegin arguments

• GL_POINTS

• GL_LINES

• GL_LINE_STRIP

• GL_LINE_LOOP

• GL_TRIANGLES

• GL_TRIANGLE_STRIP

• GL_QUAD_STRIP

• GL_TRIANGLE_FAN

Next Monday

Text Color and an Introduction to Viewing