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Mitch Richling: Burning Ship Fractal

Author: Mitch Richling
Updated: 2024-10-31

horizon_50.png

Table of Contents

1. Introduction

The Burning Ship Fractal is a bit of a Mandelbrot'alike in that its generating function is a minor modification of the one used for the Mandelbrot Set:

(1)f(z)=z2+cMandelbrot(2)f(z)=(|ℜ(z)|+i⋅|ℑ(z)|)2+cBurningShip

Granted, these two functions are not visually similar in the standard form written above; however, we can remedy that by rewriting the equations using explicit real and imaginary parts. We use x+yi instead of z for the following two equations:

(3)f(z)=(x+yi)2+cMandelbrot(4)f(z)=(|x|+|y|i)2+cBurningShip

So the equations are identical except for the signs of the real and imaginary parts – i.e. at each step in the Burning Ship Fractal we make the parts positive.

2. Algorithms

We render the Burning Ship Fractal using the L function just like we did with the Mandelbrot Set. The general idea is to map an array of pixels making up the raster image to a rectangular region of the complex plane, and color each pixel of the image based upon a color scale for the computed value of the L function. Coding up this idea leads to the following C++ program: 

#include "ramCanvas.hpp"

typedef mjr::color3c8b::csCC_tpl<mjr::color3c8b::cornerColorEnum::BLUE, 
                                 mjr::color3c8b::cornerColorEnum::RED, 
                                 mjr::color3c8b::cornerColorEnum::YELLOW, 
                                 mjr::color3c8b::cornerColorEnum::WHITE>  csCCbs;

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int NUMITR = 1024; 
  const int MAXMAG = 15; 
  const int IMGSIZ = 7680/4;
  //mjr::ramCanvas3c8b escRamCanvas(IMGSIZ, IMGSIZ, -1.5, 2.7, -1.25, 2.49);                 // Entire fractal
  //mjr::ramCanvas3c8b escRamCanvas(IMGSIZ, IMGSIZ, 1.6, 1.9, -0.089, 0.2);                  // The horizon zoom
  mjr::ramCanvas3c8b escRamCanvas(IMGSIZ, IMGSIZ, 1.715, 1.795, -0.022, 0.086);              // The classic zoom
  //mjr::ramCanvas3c8b escRamCanvas(IMGSIZ, IMGSIZ, 1.802, 1.80287, -0.002, 0.002);          // Columns to heaven
  //mjr::ramCanvas3c8b escRamCanvas(IMGSIZ, IMGSIZ, -0.3743, -0.3735, -0.0876, -0.0866);     // Deep zoom.  BATS!

  for(int y=0;y<escRamCanvas.getNumPixY();y++) {
    for(int x=0;x<escRamCanvas.getNumPixX();x++) {
      double cx = escRamCanvas.int2realX(x), cy = escRamCanvas.int2realY(y);
      double zx = 0.0, zy = 0.0;
      double t;
      int count = 0;
      while(((zx*zx+zy*zy)<MAXMAG) && (count<=NUMITR)) {
        count++;
        t  = zx*zx - zy*zy     - cx;
        zy = std::abs(2*zx*zy) - cy;
        zx = t;
      }
      if(count < NUMITR) 
        escRamCanvas.drawPoint(x, y, csCCbs::c(static_cast<mjr::ramCanvas3c8b::csIntType>(count*10)));
    }
  }
  // Escape Time Coloring
  escRamCanvas.writeTIFFfile("BurningShip.tiff");
  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

The above program will generate this picture:

BurningShip_03_50.png

4. References

Check out the fractals section of my reading list.

All the code used to generate everything on this page may be found on github.