Mitch Richling: Tricorn Fractal

Author:	Mitch Richling
Updated:	2024-10-31

1. Introduction
2. Algorithms
3. Gallery
4. References

1. Introduction

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

$\begin{array}{rclr} (1) & f (z) & = & z^{2} + c & Mandelbrot \\ (2) & f (z) & = & {\bar{z}}^{2} + c & Tricorn \end{array}$

We render Tricorn Fractals 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.

2. Algorithms

#include "ramCanvas.hpp"

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    NUMITR = 1024;
  const int    IMXSIZ = 7680/2;
  const int    IMYSIZ = 7680/2;

  mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -2.75, 2.75, -2.75, 2.75);      // The whole thing
  //mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -2.1, -0.30, -0.25, 0.25);      // The "corn"
  //mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -1.65, -1.46, -0.07, 0.07);  // Tiny tricorns & mandelbrots
  //mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -1.48-0.0065, -1.48+0.007, -0.0035, 0.0035); // That biggest mandelbrot-like set from above

  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)<14) && (count<=NUMITR)) {
        z=std::pow(std::conj(z), 2) + c;
        count++;
      }
      if(count < NUMITR)
        theRamCanvas.drawPoint(x, y, mjr::ramCanvas3c8b::colorType::csCColdeFireRamp::c(static_cast<mjr::ramCanvas3c8b::csIntType>(count*30)));
    }
  }
  theRamCanvas.writeTIFFfile("tricorn.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 the following image:

3. Gallery

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.

Mitch Richling: Tricorn Fractal

Table of Contents

1. Introduction

2. Algorithms

3. Gallery

4. References