Download Mandelbrot and Julian sets

Mandelbrot and Julian sets
Vaclav Vavra
Definitions
• By both sets we compute the same sequence
given by recursive formula:
zn = (zn-1)2 + c
• zn , zn-1, c are complex numbers(!)
• Julian set is set of z0’s for which the sequence
does not diverge
– (c is constant for a given Julian set)
• formally:
J = C - {z0| zn = (zn-1)2 + c → ∞}
J = {z0| ┐(zn = (zn-1)2 + c → ∞)}
or
Definitions
• Mandelbrot set is set of c’s for which the
sequence does not diverge for z0 = 0
• formally:
J = C - {c|z0 = 0, zn = (zn-1)2 + c → ∞}
J = {c|z0 = 0, ┐(zn = (zn-1)2 + c → ∞)}
or
Operations in C
• Notation:
x = a + b.i
– a is real part, b is imaginary part, i is
imaginary unit, i2=-1
• Operations:
y = c + d.i
x + y = (a+c) + (b+d).i
x.y = (ac−bd) + (bc+ad).i
|x| = sqrt(a2 + b2)
Visualisation
• for a given c, z0 we compute z1, z2, z3,…in
a loop
• this way we cannot analytically determine,
whether the sequence diverges
• However if |zn| > 2, it really diverges
• if |zn| is still <= 2, we just stop after fixed
number of iterations
• It works
Julian set – sudocode
const complex c = {c.r,c.i};
for (every real part for z0) {
for (every imaginary part for z0) {
//should be between -2 and 2
//should be between -2 and 2
z := z0 ;
for (fixed number of iterations) {
z = z^2 + c;
if (|z| > 2) break;
}
//120 is ok
// or |z|^2 > 4
drawPixel(Re(z0), Im(z0), #iterations needed);
}
}
Mandelbrot set – sudocode
for (every real part for c) {
for (every imaginary part for c) {
z := c;
//should be between -2 and 2
//should be between -2 and 2
//z0=0, therefore z1=c
for (fixed number of iterations) {
z = z^2 + c;
if (|z| > 2) break;
}
//40 is ok here
// or |z|^2 > 4
drawPixel(Re(z0), Im(z0), #iterations needed);
}
}
Colors
•
•
•
•
•
We map #iterations to colors
Various ways how to do it
Besides #iterations #maximum number of iterations (maxiter)
We want to set values for r,g,b
Maybe you want point inside the set to be black
– If (#iterations == maxiter) r = b = g = 0;
Examples:
a) (r,g,b are floats from 0.0 to 1.0)
r = 1-(1-#iterations/maxiter)^5;
g = 1-(1-#iterations/maxiter)^3;
b = #iterations/maxiter;
Colors
b) (r,g,b are integers from 0 to 255)
int colorTable[16][3] = {
{ 0, 0, 0},
{ 0, 0,170},
{ 0,170, 0},
{ 0,170,170},
{170, 0, 0},
{170, 0,170},
{170, 85, 0},
{170,170,170},
{ 85, 85, 85},
{ 85, 85,255},
{ 85,255, 85},
{ 85,255,255},
{255, 85, 85},
{255, 85,255},
{255,255, 85},
{255,255,255},
};
r = colorTable[#iterations % 16][0];
g = colorTable[#iterations % 16][1];
b = colorTable[#iterations % 16][2];
You can find other examples at http://www.root.cz/clanky/fraktaly-v-pocitacove-grafice-xiv/
Tips and tricks
• colors: instead of mapping #iterations to colors, we can
map zn (the last one before we leave the loop) to colors
• So now we have
drawPixel(Re(z0), Im(z0), Re(zn), Im(zn));
– We can use it for the inside if the set too!!
• Until now we get drawPixel(Re(z0), Im(z0),maxiter) there
• Mandelbrot set – if we set z0 to a non-zero value, we get
the set deformed
– z0 is then called the “perturbation” term
• Set the maximum number of iterations to a small number
and we have the sets deformed, too (particularly useful
for Mandelbrot set)
References
• http://www.root.cz/clanky/fraktaly-v-pocitacove-grafice-xiv
• http://www.cis.ksu.edu/~vaclav/fractals.html