Mitch Richling: The Dancing Biomorph

Author:	Mitch Richling
Updated:	2024-10-31

1. Introduction
2. Algorithm & Code
3. Gallery
4. References

1. Introduction

Given some function, $f (z)$ , we define a series for each point in the complex plane, $z_{0}$ , by $g^{(n)} (z_{0})$ where $g (z) = f (z) + c$ for some fixed $c$ . This is pretty similar to how we draw the Mandelbrot Set, but note the different role played by the variable $c$ ! For any given $f (z)$ , the $c$ parameter defines a fractal – so for each function $f (z)$ we have a family of fractals.

Unlike the Mandelbrot Set, iteration stops for a biomorph when $| ℜ (z) | \geq 10$ and $| ℑ (z) | \geq 10$ .

Another difference between biomorphs and the Mandelbrot Set is the way they are normally colorized. Biomorphs are generally colored based on the escape function, but by a combination of escape function, magnitude/phase of z at iteration end, etc…

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 = 7;
  const int    NUMFRM = 24*4;
  const int    IMXSIZ = 7680/4;
  const int    IMYSIZ = 7680/4;
  const int    LIM    = 10;

  mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -2.75, 2.75, -2.75, 2.75);
  for(int frame=0; frame<NUMFRM; frame++) {
    std::cout << "Frame: " << frame << std::endl;
    double angle = frame*std::numbers::pi*2/NUMFRM;    
    theRamCanvas.clrCanvasToBlack();
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> c(std::cos(angle), std::sin(angle));
        std::complex<double> z = theRamCanvas.int2real(x, y);
        std::complex<double> zL(0.0, 0.0);
        int count = 0; 
        while( ((std::abs(std::real(z))<LIM) || (std::abs(std::imag(z))<LIM)) && (count<NUMITR) ) {
          zL = z;
          z=std::sin(z) + std::pow(z, 2) + c;
          count++;
        }
        if(count < NUMITR) {
          if(std::abs(std::real(zL)) < LIM)
            theRamCanvas.drawPoint(x, y, ct::csCCu0W::c(std::abs(std::real(zL))/LIM));
          else if(std::abs(std::imag(zL)) < LIM)
            theRamCanvas.drawPoint(x, y, ct::csCCu0W::c(1.0-std::abs(std::imag(zL))/LIM));
        } else {
          if(std::abs(std::real(zL)) < LIM)
            theRamCanvas.drawPoint(x, y, ct::csCCu0Y::c(std::abs(std::real(zL))/LIM));
          else if(std::abs(std::imag(zL)) < LIM)
            theRamCanvas.drawPoint(x, y, ct::csCCu0Y::c(1.0-std::abs(std::imag(zL))/LIM));
        }
      }
    }
    theRamCanvas.writeTIFFfile("biomorphMorph_" + 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;
}