Mitch Richling: Mandelbrot Wave

Author:	Mitch Richling
Updated:	2024-10-31

1. Introduction
2. Algorithm & Code
3. References

1. Introduction

Iterate \(f(x)=z^5+a\cdot z+c\) as one would \(f(z)=z^2+c\) to render a Mandelbrot Set. Vary the parameter \(a\) by moving it around the unit circle, and drop out an image for each \(a\).

2. Algorithm & Code

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;
typedef ct::csIntType cit;

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    NUMITR = 255;
  const int    NUMFRM = 24*16;      // Full circle animation
  //const int    NUMFRM = 24*4;     // Wave animation
  const int    IMXSIZ = 7680/2;
  const int    IMYSIZ = 7680/2;
  const int    MAXZSQ = 10;
  const double ANGMIN = 0.0;                 // Full circle animation
  const double ANGMAX = std::numbers::pi*2;  // Full circle animation
  // const double ANGMIN = 3.04341788317;    // Wave animation
  // const double ANGMAX = 3.23976742401;    // Wave animation

  for(int frame=0; frame<NUMFRM; frame++) {
    mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -1.5, 1.5, -1.5, 1.5);    // Full circle animation
    // mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -0.7, 0.1, -0.7, 0.1); // Wave animation
    theRamCanvas.clrCanvasToBlack();
    std::cout << "Frame: " << frame << " of " << NUMFRM << std::endl;
    double angle = frame*(ANGMAX-ANGMIN)/NUMFRM+ANGMIN;    
    std::complex<double> a(std::cos(angle), std::sin(angle));
#   pragma omp parallel for schedule(static,1)
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> c(theRamCanvas.int2realX(x), theRamCanvas.int2realY(y));
        std::complex<double> z(0.0, 0.0);
        int count = 0; 
        while((std::norm(z)<MAXZSQ) && (count<=NUMITR)) {
          z = std::pow(z, 5) +  a * z + c;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, ct::csCColdeFireRamp::c(static_cast<ct::csIntType>(count*20)));
      }
    }
    theRamCanvas.scaleDownMean(8);   // Another good choice: theRamCanvas.scaleDownMax(8); 
    theRamCanvas.writeTIFFfile("mandelbrot_wave_" + mjr::math::str::fmt_int(frame, 3, '0') + ".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 frames for a movie which may be rendered into a GIF:

If we use the scaleDownMax option, then we get the following:

Lastly, if we use the "wave" code, we get the following:

3. 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.

Mitch Richling: Mandelbrot Wave

Table of Contents

1. Introduction

2. Algorithm & Code

3. References