1. Computer Graphics - Lecture 6
Fractal Graphics
Objectives
To give a definition and a classification of fractals.
To present some other deterministic self-similar fractal curves.
To introduce the most ‘important’ fractals: Julia and Mandelbrot.
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Computer Graphics - Lecture 6

2. Computer Graphics - Lecture 6
Fractals
A fractal is a set of points such that:
- its fractal dimension is infinite [infinite detail at every point].
- satisfies self-similarity: any part of the fractal is similar with the fractal.
Generating a fractal is a iterative process:
- start from P0
- iteratively generate P1=F(P0), P2=F(P1), …, Pn=F(Pn-1), …
P0 is a set of initial points
F is a transformation:
Geometric transformations: translations, rotations, scaling, …
Non-Linear coordinate transformation.
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Computer Graphics - Lecture 6

3. Computer Graphics - Lecture 6
Clasification
The nature of F gives:
Deterministic Fractals:
Stochastic Fractals: F is a random process.
The aspect of the fractal set:
Self-Similar: parts are scaled down of a fractal pattern.
Self-Affine: formed by scaling with different parameters.
Non-Linear Fractals: formed by non-linear transformation.
The most popular fractals:
- Koch’s fractal or Snowflake.
- The Julia sets.
- The Mandelbrot set.
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Computer Graphics - Lecture 6

4. Computer Graphics - Lecture 6
Koch’s curve
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5. Computer Graphics - Lecture 6
Julia Set
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6. Computer Graphics - Lecture 6
Mandelbrot Set
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Computer Graphics - Lecture 6

7. Deterministic Self-Similar Fractals
Deterministic Self-Similar Fractals = Geometric Fractals.
Similarity gives a generator pattern.
Based on GP we can find the transformation F.
Example:Quadratic Koch’s curve 
K(1,l) is a line
K(n,l)= K(n-1,l/4);right(90);K(n-1,l/4);left(90);K(n-1,l/4);left(90);
K(n-1,n/4); K(n-1,n/4);right(90); K(n-1,n/4);right(90); K(n-1,n/4);
left(90); K(n-1,n/4)
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Computer Graphics - Lecture 6

8. Quadratic Koch’s Curve
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9. Computer Graphics - Lecture 6
Complex Numbers
C the set complex numbers z=a+j*b, a,bR and j2=-1
Operations with Complex Numbers:
z1 + z2 = (a1+j*b1) + (a2+j*b2)=(a1 + a2 ) +j*(b1+ b2).
z1 * z2 = (a1+j*b1) * (a2+j*b2)=(a1*a2 - b1*b2 ) +j*(a1*b2 + b1*a2 ).
| z1 | = sqrt(a12+ b12)
b z=a+i*b
|z| a
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Computer Graphics - Lecture 6

10. Computer Graphics - Lecture 6
Complex Numbers (2)
Complex Numbers in trigonometrically format:
z=a+j*b=|z|(cos +j*sin )
z1 * z2 = |z1|(cos1+j*sin1)*|z2|(cos2+j*sin 2)=
=|z1* z2 | (cos(1+ 2)+j*sin(1+ 2))
b z=a+i*b
|z| a 
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Computer Graphics - Lecture 6

11. Computer Graphics - Lecture 6
Complex Class
public class Complex{
private double re, im;
public Complex(double x0,double y0) {re=x0;im=y0;}
public Complex(){re=im=0;}
public Complex(Complex z){re=z.re;im=z.im;}
public double getRe(){return re;}
public double getIm(){return im;}
public double getAbs(){return Math.sqrt(re*re+im*im);}
public Complex conjugate(){return new Complex(re,-im);}
public static Complex sum(Complex z1,Complex z2){
Complex sum = new Complex();
sum.re=z1.re+z2.re;sum.im=z1.im+z2.im;
return sum; }
public static Complex product(Complex z1,Complex z2){
Complex prod = new Complex();
prod.re=z1.re*z2.re-z1.im*z2.im;
prod.im=z1.im*z2.re+z2.im*z1.re;
return prod; }
public void write()
{System.out.println("" + re + " +i* " + im);}
public static Complex read(){
Complex z = new Complex();
System.out.print("re = ");z.re=Read.readDouble();
System.out.print("im = ");z.im=Read.readDouble();
return z;
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Computer Graphics - Lecture 6

12. Julia Sets – Self-Squaring Fractals
Consider the generating function F(z)=z2+c, z,c C.
Sequence of complex numbers: z0C and zn+1= zn2 + c.
Chaotic behaviour but two attractors for |zn|: 0 and +.
For a c C, Julia’s set Jc represents all the points whose orbit is finite.
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Computer Graphics - Lecture 6

13. Computer Graphics - Lecture 6
Julia Sets – Algorithm
Inputs:
c C the complex number; [xmin,xmax] * [ymin,ymax] a region in plane.
Niter a number of iterations for orbits; R a threshold for the attractor .
Output:
Jc the Julia set of c
Algorithm
For each (x,y) in [xmin,xmax] * [ymin,ymax]
construct z0=x+j*y;
find the orbit of z0 [first Niter elements]
if (all the orbit points are under the threshold) draw (x,y)
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14. Computer Graphics - Lecture 6
public void paint(Graphics g){
g.setColor(Color.red);g.fillRect(10,10,SIZE,SIZE);
g.setColor(Color.yellow);
for(int i=0;i<SIZE;i++)for(int j=0;j<SIZE;j++){
Complex z = new Complex(XMIN+i*STEP,YMIN+j*STEP);
int k; for (k=0;k<NR_ITER;k++){
z=f(z,c);
if(z.getAbs()>R)break; }
if (k==NR_ITER) g.drawOval (10+i,10+j,1,1);
The whole java code is L7Appl1.java and the applet is L7Appl1.html.
Some values that give nice fractals:
c=-0.83+j*0.16,  c=-0.13+j*0.76, c=-0.39+j*0.58, c=0.785+j*0.13, etc.  
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15. Computer Graphics - Lecture 6
                                                 
c = 0.275
c = 1/4
c = 0
c = -3/4
c = -1.312
c = -1.375
c = -2
c = i
c = (+0.285, +0.535)
c = (-0.125, +0.750)
c = (-0.500, +0.563)
c = (-0.687, +0.312)
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16. Computer Graphics - Lecture 6
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17. Computer Graphics - Lecture 6
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18. Computer Graphics - Lecture 6
Maldelbrot Set
Maldelbrot Set contains all the points cC such that
z0=0 and zn+1= zn2 + c has an finite orbit.
Inputs: [xmin,xmax] * [ymin,ymax] a region in plane.
Niter a number of iterations for orbits; R a threshold for the attractor .
Output:
M the Mandelbrot set.
Algorithm
For each (x,y) in [xmin,xmax] * [ymin,ymax]
c=x+i*y;
find the orbit of z0=0 while under the threshold.
if (all the orbit points are not under the threshold) draw c(x,y)
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Computer Graphics - Lecture 6

19. Computer Graphics - Lecture 6
public void paint(Graphics g){
g.setColor(Color.blue);g.fillRect(10,10,SIZE,SIZE);
for(int i=0;i<SIZE;i++)for(int j=0;j<SIZE;j++){
Complex c = new Complex(XMIN+i*STEP,YMIN+j*STEP); // create a number c
Complex z = new Complex(); // create z=0
int k;
// find the orbit
for (k=0;k<NR_ITER;k++){
z=f(z,c);
// the point is above the threshold
if(z.getAbs()>R){
g.setColor(col[k%col.length]);
g.drawOval(10+i,10+j,1,1);
break; }
The whole java file L7Appl2.java and the applet is L7Appl2.html.
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20. Computer Graphics - Lecture 6
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21. Computer Graphics - Lecture 6
Problems to Solve
1. Draw the self-similar fractal with the generator 
2. Modify the function F(z)=z2+c and find some new fractal sets.
F(z)= - z2+c, F(z)=z*(1-z)+c of even F(z)=z3+c.
To Read:
Peter Cooley, The Essence of Computer Graphics, (Modelling Natural Objects), pp75-93.
Some web links: (
4/25/2019
Computer Graphics - Lecture 6