Smooth spectrum for Mandelbrot Set rendering

Question

I\'m currently writing a program to generate really enormous (65536x65536 pixels and above) Mandelbrot images, and I\'d like to devise a spectrum and coloring scheme that do

Answer 1

This is the smooth color algorithm:

Lets say you start with the complex number z0 and iterate n times until it escapes. Let the end point be zn.

A smooth value would be

nsmooth := n + 1 - Math.log(Math.log(zn.abs()))/Math.log(2)

This only works for mandelbrot, if you want to compute a smooth function for julia sets, then use

Complex z = new Complex(x,y);
double smoothcolor = Math.exp(-z.abs());

for(i=0;i

Then smoothcolor is in the interval (0,max_iter).

Divide smoothcolor with max_iter to get a value between 0 and 1.

To get a smooth color from the value:

This can be called, for example (in Java):

Color.HSBtoRGB(0.95f + 10 * smoothcolor ,0.6f,1.0f);

since the first value in HSB color parameters is used to define the color from the color circle.