Just Some Julia Sets

So one of the neat side things I learned at GHP was how the Mandelbrot Set works. Somehow I had the impression that the Mandelbrot Set was really complicated so I had never looked up the details. But it is really crazy-simple, assuming you know what a complex number is. Anyways, once I learned this, I just had to cook up my own mini-fractal generator and explore. I didn’t actually draw the Mandelbrot (yeah, now that I understand the Mandelbrot just doesn’t excite me anymore): I’m drawing some julia sets which are a very close relative.

And I did it in Processing which made it extra easy and fun.

[sourcecode language=”java” collapse=”true”] class Cnum //complex number {

float r; //real
float i; //imaginary
Cnum(float r_, float i_) {
    r = r_;
    i = i_;
}

String irep()
{
    return "[" + r + " + " + i + "i]";
}

String prep()
{
    float theta = arg();
    float r2 = sqrt((i*i) + (r*r));
    return "[r=" + r2 + " theta=" + theta + "]";

}

String toString() {
    return irep();
}

boolean isInfinity()
{
    return r == Float.POSITIVE_INFINITY;
}

boolean isZero()
{
    return r == 0 && i == 0;
}

Cnum conj() 
{
    return new Cnum(r, i*-1);
}

Cnum neg()
{
    return new Cnum(-r,-i);
}


float arg()
{
return atan2(i,r);
}

float abs()
{
    return sqrt(r*r + i*i);
} } 	

Cnum add(Cnum a, Cnum b) { return new Cnum(a.r + b.r, a.i + b.i); } Cnum sub(Cnum a, Cnum b) { return new Cnum(a.r - b.r, a.i - b.i); }

Cnum mul(Cnum a, Cnum b) { return new Cnum(a.rb.r - a.ib.i, a.ib.r + a.rb.i); }

int WIDTH = 500; int HEIGHT = 500; boolean changeOnMove = true;

void plot(Cnum a) { float xPercentage = (a.r - upperLeft.r)/(lowerRight.r - upperLeft.r); float yPercentage = (a.i - lowerRight.i)/(upperLeft.i - lowerRight.i); int x = round(xPercentageWIDTH); int y = round((1 - yPercentage)HEIGHT); point(x, y); }

Cnum convert(int x, int y) { float xPercentage = x/float(WIDTH); float yPercentage = y/float(HEIGHT); return new Cnum((lowerRight.r - upperLeft.r)xPercentage + upperLeft.r, (upperLeft.i - lowerRight.i)(1 - yPercentage) + lowerRight.i); }

Cnum upperLeft, lowerRight;

PFont fontA;

void setup() { upperLeft = new Cnum(-2, 2); lowerRight = new Cnum(2, -2); size(WIDTH, HEIGHT); stroke(226);

fill(226);

fontA = loadFont(“SansSerif.plain-12.vlw”);

// Set the font and its size (in units of pixels) textFont(fontA, 12); oldMouse = new Cnum(0,0);

}

void keyPressed() { if (key == ‘q’) exit(); if (key == ‘s’) save(“output.png”); if (key == ‘ ‘) { changeOnMove = !(changeOnMove); draggedX = 0; draggedY = 0; }

}

Cnum oldMouse; int maxIter = 20; void draw() { Cnum mouse; background(0);

if(changeOnMove) {
mouse = convert(mouseX, mouseY);
if(mouse.r == oldMouse.r && mouse.i == oldMouse.i) {
    if(maxIter < 1000)
	maxIter = maxIter + 20;
    else 
	return;
} else {
    maxIter = 5;
    oldMouse = mouse;
}
} else {
mouse = oldMouse;
}
for(int q = 0; q < WIDTH; q++)
for(int j = 0; j < HEIGHT; j++) {
    Cnum orig = convert(q,j);
    //Cnum acc = add(mul(orig,orig), mouse) ;
    Cnum acc = orig ;
    //Cnum acc = add(orig, mouse);
    for(int i = 0; i < maxIter; i++) {
	acc = add(mul(mul(acc,acc),acc), mouse) ;
	if(acc.abs() > 2){
	    int strokeColor = 255 - i*5;
	    if (strokeColor < 0 ) strokeColor = 0;
	    stroke(strokeColor);		    
	    plot(orig);
	    break;
	} 
    }
}
fill(0,0,222);		    
text("z^2 + cl c=" + mouse.r + " + " + mouse.i + "i", 30, 60);
if(draggedX != 0 || draggedY != 0) {
stroke(0,0,222);
noFill();
int max = max(mouseX - draggedX, mouseY - draggedY);
rect(draggedX, draggedY, max, max);
}

}

int draggedX = 0; int draggedY = 0;

void mousePressed() { if(!changeOnMove) { draggedX = mouseX; draggedY = mouseY; } }

void mouseReleased() { if(!changeOnMove && draggedX != 0 && draggedY != 0) {

int temp;
if(mouseX < draggedX) {
    temp = mouseX;
    mouseX = draggedX;
    draggedX = temp;
}
if(mouseY < draggedY) {
    temp = mouseY;
    mouseY = draggedY ;
    draggedY = temp;
}
int max = max (mouseX - draggedX, mouseY - draggedY);


Cnum lowerRightNew = convert(draggedX + max, draggedY + max);
Cnum upperLeftNew = convert(draggedX, draggedY);

upperLeft = upperLeftNew;
lowerRight = lowerRightNew;
draggedX = 0;
draggedY = 0;
}

} [/sourcecode]

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