Mitch Richling: Tricorn Fractal
| Author: | Mitch Richling |
| Updated: | 2024-10-31 |
Table of Contents
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:
\[\begin{eqnarray} f(z) & = & z^2+c & \,\,\,\,\,\,\mathrm{Mandelbrot} \\ f(z) & = & \bar{z}^2+c & \,\,\,\,\,\,\mathrm{Tricorn} \\ \end{eqnarray}\]
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
