Mandelbrot zoom problem

Programming Questions

asimes

in Programming Questions • 1 year ago

I changed Daniel Shiffman's learning code for Mandelbrot a little bit to be able to zoom in where I click. After a little more than a dozen clicks it fails (no error, the image fails). I'm wondering if there is a better way to do this? I'm not sure why exactly it fails:

float xmin = -2.25;
float ymin = -1.5;
float wh = 3;
int maxIterations = 255;
void setup() {
size(400, 400, P2D);
}
void draw() {
loadPixels();
float xmax = xmin+wh;
float ymax = ymin+wh;
float dx = (xmax-xmin)/width;
float dy = (ymax-ymin)/height;
float cx = xmin;
for (int i = 0; i < width; i++) {
float cy = ymin;
for (int j = 0; j < height; j++) {
float zx = 0;
float zy = 0;
int numIterations = 0;
while (numIterations < maxIterations) {
float zzx = zx*zx;
float zzy = zy*zy;
float twoComplex = 2.0*zx*zy;
zx = zzx-zzy+cx;
zy = twoComplex+cy;
if (zx+zy > 16.0) break;
numIterations++;
}
if (numIterations == maxIterations) pixels[i+j*width] = 0;
else pixels[i+j*width] = numIterations+(numIterations<<8)+(numIterations<<16);
cy += dy;
}
cx += dx;
}
updatePixels();
noLoop();
}
void mousePressed() {
wh /= 2;
float xmax = xmin+wh;
float ymax = ymin+wh;
float dx = (xmax-xmin)/width;
float dy = (ymax-ymin)/height;
xmin += dx*mouseX;
ymin += dy*mouseY;
loop();
}

Replies(3)

jbum

Re: Mandelbrot zoom problem

1 year ago

Two things:

1) You are going to want to start increasing the value of maxIterations as you zoom in. The value of 256 creates a limit to the visible detail that becomes apparent as you zoom in.

2) There is a limit to how far you can zoom in before the pixels will start to look 'blocky'. This is caused by finite resolution due to the nature of the 'float'. You can go a few more clicks if you change your floats to doubles. To go much further in, you'll have to use a class for doing arithmetic with big numbers (e.g. BigNums), and this will slow rendering to a crawl.

asimes

Re: Mandelbrot zoom problem

1 year ago

...one more question on this, I hope I'm not driving you crazy at this point. I made a method of antialiasing the image which helped make it look much better (although at the cost of speed obviously). Another problem I'm getting are annoying bands of color. The wikipedia article says that this is part of the "Escape Time" algorithm which seems to be what Shiffman used:

http://en.wikipedia.org/wiki/Mandelbrot_set#Escape_time_algorithm

I'm trying to figure out how the continuous smooth color works to get rid of them but the explanation is beyond my equation reading abilities:

http://en.wikipedia.org/wiki/Mandelbrot_set#Continuous_.28smooth.29_coloring

Any help would be appreciated:

float xmin = -2.5;
float ymin = -2.0;
float wh = 4;
int antiAlias = 4;
int maxIterations = 1000;
void setup() {
size(400, 400, P2D);
}
void draw() {
loadPixels();
float xmax = xmin+wh;
float ymax = ymin+wh;
float dx = (xmax-xmin)/(width*antiAlias);
float dy = (ymax-ymin)/(height*antiAlias);
float cx = xmin;
int[] nums = new int[width*antiAlias*height*antiAlias];
for (int i = 0; i < width*antiAlias; i++) {
float cy = ymin;
for (int j = 0; j < height*antiAlias; j++) {
float zx = 0;
float zy = 0;
int numIterations = 0;
while (numIterations < maxIterations) {
float zzx = zx*zx;
float zzy = zy*zy;
float twoComplex = 2.0*zx*zy;
zx = zzx-zzy+cx;
zy = twoComplex+cy;
if (zx+zy > 16.0) break;
numIterations++;
}
if (numIterations == maxIterations) nums[i+(j*width*antiAlias)] = 0;
else nums[i+(j*width*antiAlias)] = numIterations;
cy += dy;
}
cx += dx;
}
for (int i = 0; i < width*antiAlias; i += antiAlias) {
for (int j = 0; j < height*antiAlias; j += antiAlias) {
float total = 0;
for (int k = 0; k < antiAlias; k++) {
for (int l = 0; l < antiAlias; l++) {
total += nums[i+j*width*antiAlias+k+l*width*antiAlias];
}
}
total /= antiAlias*antiAlias;
pixels[i/antiAlias+(j*width)/antiAlias] = color(total*16%255);
}
}
updatePixels();
noLoop();
}
void mousePressed() {
wh /= 2;
float xmax = xmin+wh;
float ymax = ymin+wh;
float dx = (xmax-xmin)/(width*antiAlias);
float dy = (ymax-ymin)/(height*antiAlias);
xmin += dx*mouseX*antiAlias;
ymin += dy*mouseY*antiAlias;
loop();
}

jbum

Re: Mandelbrot zoom problem

1 year ago

There are some (likely out of print) books that provide pseudo source for a few smooth coloring algorithms. One that comes to mind is "The Beauty of Fractals" by Pietgen.

Another trick you might want to try is progressive quad tree rendering, which will enable you to get an overall picture before filling in the details. Helpful for quickly navigating around the space. I wrote a pretty decent Mandelbrot explorer in C++ some years back, that uses OpenGL...

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