Mandelbrot.cs

using System;
using System.Drawing;

namespace Algorithms.Other;

/// <summary>
///     The Mandelbrot set is the set of complex numbers "c" for which the series
///     "z_(n+1) = z_n * z_n + c" does not diverge, i.e. remains bounded. Thus, a
///     complex number "c" is a member of the Mandelbrot set if, when starting with
///     "z_0 = 0" and applying the iteration repeatedly, the absolute value of
///     "z_n" remains bounded for all "n > 0". Complex numbers can be written as
///     "a + b*i": "a" is the real component, usually drawn on the x-axis, and "b*i"
///     is the imaginary component, usually drawn on the y-axis. Most visualizations
///     of the Mandelbrot set use a color-coding to indicate after how many steps in
///     the series the numbers outside the set cross the divergence threshold.
///     Images of the Mandelbrot set exhibit an elaborate and infinitely
///     complicated boundary that reveals progressively ever-finer recursive detail
///     at increasing magnifications, making the boundary of the Mandelbrot set a
///     fractal curve.
///     (description adapted from https://en.wikipedia.org/wiki/Mandelbrot_set)
///     (see also https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set).
/// </summary>
public static class Mandelbrot
{
    /// <summary>
    ///     Method to generate the bitmap of the Mandelbrot set. Two types of coordinates
    ///     are used: bitmap-coordinates that refer to the pixels and figure-coordinates
    ///     that refer to the complex numbers inside and outside the Mandelbrot set. The
    ///     figure-coordinates in the arguments of this method determine which section
    ///     of the Mandelbrot set is viewed. The main area of the Mandelbrot set is
    ///     roughly between "-1.5 &lt; x &lt; 0.5" and "-1 &lt; y &lt; 1" in the figure-coordinates.
    ///     To save the bitmap the command 'GetBitmap().Save("Mandelbrot.png")' can be used.
    /// </summary>
    /// <param name="bitmapWidth">The width of the rendered bitmap.</param>
    /// <param name="bitmapHeight">The height of the rendered bitmap.</param>
    /// <param name="figureCenterX">The x-coordinate of the center of the figure.</param>
    /// <param name="figureCenterY">The y-coordinate of the center of the figure.</param>
    /// <param name="figureWidth">The width of the figure.</param>
    /// <param name="maxStep">Maximum number of steps to check for divergent behavior.</param>
    /// <param name="useDistanceColorCoding">Render in color or black and white.</param>
    /// <returns>The bitmap of the rendered Mandelbrot set.</returns>
    public static Bitmap GetBitmap(
        int bitmapWidth = 800,
        int bitmapHeight = 600,
        double figureCenterX = -0.6,
        double figureCenterY = 0,
        double figureWidth = 3.2,
        int maxStep = 50,
        bool useDistanceColorCoding = true)
    {
        if (bitmapWidth <= 0)
        {
            throw new ArgumentOutOfRangeException(
                nameof(bitmapWidth),
                $"{nameof(bitmapWidth)} should be greater than zero");
        }

        if (bitmapHeight <= 0)
        {
            throw new ArgumentOutOfRangeException(
                nameof(bitmapHeight),
                $"{nameof(bitmapHeight)} should be greater than zero");
        }

        if (maxStep <= 0)
        {
            throw new ArgumentOutOfRangeException(
                nameof(maxStep),
                $"{nameof(maxStep)} should be greater than zero");
        }

        var bitmap = new Bitmap(bitmapWidth, bitmapHeight);
        var figureHeight = figureWidth / bitmapWidth * bitmapHeight;

        // loop through the bitmap-coordinates
        for (var bitmapX = 0; bitmapX < bitmapWidth; bitmapX++)
        {
            for (var bitmapY = 0; bitmapY < bitmapHeight; bitmapY++)
            {
                // determine the figure-coordinates based on the bitmap-coordinates
                var figureX = figureCenterX + ((double)bitmapX / bitmapWidth - 0.5) * figureWidth;
                var figureY = figureCenterY + ((double)bitmapY / bitmapHeight - 0.5) * figureHeight;

                var distance = GetDistance(figureX, figureY, maxStep);

                // color the corresponding pixel based on the selected coloring-function
                bitmap.SetPixel(
                    bitmapX,
                    bitmapY,
                    useDistanceColorCoding ? ColorCodedColorMap(distance) : BlackAndWhiteColorMap(distance));
            }
        }

        return bitmap;
    }

    /// <summary>
    ///     Black and white color-coding that ignores the relative distance. The Mandelbrot
    ///     set is black, everything else is white.
    /// </summary>
    /// <param name="distance">Distance until divergence threshold.</param>
    /// <returns>The color corresponding to the distance.</returns>
    private static Color BlackAndWhiteColorMap(double distance) =>
        distance >= 1
            ? Color.FromArgb(255, 0, 0, 0)
            : Color.FromArgb(255, 255, 255, 255);

    /// <summary>
    ///     Color-coding taking the relative distance into account. The Mandelbrot set
    ///     is black.
    /// </summary>
    /// <param name="distance">Distance until divergence threshold.</param>
    /// <returns>The color corresponding to the distance.</returns>
    private static Color ColorCodedColorMap(double distance)
    {
        if (distance >= 1)
        {
            return Color.FromArgb(255, 0, 0, 0);
        }

        // simplified transformation of HSV to RGB
        // distance determines hue
        var hue = 360 * distance;
        double saturation = 1;
        double val = 255;
        var hi = (int)Math.Floor(hue / 60) % 6;
        var f = hue / 60 - Math.Floor(hue / 60);

        var v = (int)val;
        var p = 0;
        var q = (int)(val * (1 - f * saturation));
        var t = (int)(val * (1 - (1 - f) * saturation));

        switch (hi)
        {
            case 0: return Color.FromArgb(255, v, t, p);
            case 1: return Color.FromArgb(255, q, v, p);
            case 2: return Color.FromArgb(255, p, v, t);
            case 3: return Color.FromArgb(255, p, q, v);
            case 4: return Color.FromArgb(255, t, p, v);
            default: return Color.FromArgb(255, v, p, q);
        }
    }

    /// <summary>
    ///     Return the relative distance (ratio of steps taken to maxStep) after which the complex number
    ///     constituted by this x-y-pair diverges. Members of the Mandelbrot set do not
    ///     diverge so their distance is 1.
    /// </summary>
    /// <param name="figureX">The x-coordinate within the figure.</param>
    /// <param name="figureY">The y-coordinate within the figure.</param>
    /// <param name="maxStep">Maximum number of steps to check for divergent behavior.</param>
    /// <returns>The relative distance as the ratio of steps taken to maxStep.</returns>
    private static double GetDistance(double figureX, double figureY, int maxStep)
    {
        var a = figureX;
        var b = figureY;
        var currentStep = 0;
        for (var step = 0; step < maxStep; step++)
        {
            currentStep = step;
            var aNew = a * a - b * b + figureX;
            b = 2 * a * b + figureY;
            a = aNew;

            // divergence happens for all complex number with an absolute value
            // greater than 4 (= divergence threshold)
            if (a * a + b * b > 4)
            {
                break;
            }
        }

        return (double)currentStep / (maxStep - 1);
    }
}