Famous 2D Mandelbrot Equation

/*
Minimal working (Mandelbulb xbdev.net)

PPM image format is a simple and uncomplicated way to generate an image without
requiring any external libraries.  Easy to open and convert to another format
(e.g., open and save as jpg using paint/xnview).
*/

#define _CRT_SECURE_NO_WARNINGS

// constants to tweak and control the fractal output
#define BAILOUT        100
#define EPS            0.001
#define mmaxIterations 10

#define maxDetailIter  100
// e.g., try and change this value to 5 instead
// and see what the generated output looks like
#define myPower        12

#include <math.h>
#include <iostream>
#include <stdlib.h>
#include <stdio.h>

using namespace std;

#define M_PI 3.14159265358979323846264338327950288

inline double clamp(double x) { return x < 0 ? 0 : x>1 ? 1 : x; }
inline double min(double x, double y){ if (x<y) return x; return y; }
inline double max(double x, double y){ if (x>y) return x; return y; }
inline int toInt(double x) { return int(pow(clamp(x), 1 / 2.2) * 255 + .5); }
inline double clamp(const double ff, double lo, double hi)
{
    if (ff > hi) return hi;
    if (ff < lo) return lo;
    return ff;
}

struct vec3 {
    double x, y, z;                  // position, also color (r,g,b)
    vec3(double x_ = 0, double y_ = 0, double z_ = 0) { x = x_; y = y_; z = z_; }
    vec3 operator+(const vec3 &b) const { return vec3(x + b.x, y + b.y, z + b.z); }
    vec3 operator-(const vec3 &b) const { return vec3(x - b.x, y - b.y, z - b.z); }
    vec3 operator*(double b) const { return vec3(x*b, y*b, z*b); }
    vec3 mult(const vec3 &b) const { return vec3(x*b.x, y*b.y, z*b.z); }
    vec3& norm() { return *this = *this * (1 / sqrt(x*x + y*y + z*z)); }
    double dot( const vec3 &b) const { return x*b.x + y*b.y + z*b.z; }
    vec3 cross (vec3&b) { return vec3(y*b.z - z*b.y, z*b.x - x*b.z, x*b.y - y*b.x); }
};

struct vec4 {
    double x, y, z, w;
    vec4(const vec3& v, double _w){ x=v.x, y=v.y, z=v.z, w=_w; }
    vec4(double x_ = 0, double y_ = 0, double z_ = 0, double w_ = 0) { x = x_; y = y_; z = z_; w = w_; }
    vec4 operator+(const vec4 &b) const { return vec4(x + b.x, y + b.y, z + b.z, w + b.w); }
    vec4 operator-(const vec4 &b) const { return vec4(x - b.x, y - b.y, z - b.z, w - b.w); }
    vec4 operator*(double b) const { return vec4(x*b, y*b, z*b, w*b); }
    vec4 mult(const vec4 &b) const { return vec4(x*b.x, y*b.y, z*b.z, w*b.w); }
    vec4& norm() { return *this = *this * (1 / sqrt(x*x + y*y + z*z + w*w)); }
    double dot(const vec4 &b) const { return x*b.x + y*b.y + z*b.z, w*b.w; }

};

double dot(const vec3&a, const vec3 &b) { return a.dot(b); }
double length(const vec3& v) { return sqrt(v.dot(v)); }
double length(const vec4& v) { return sqrt(v.dot(v)); }
vec3 normalize(const vec3& v) {
    double len = length(v);
    return vec3( v.x/len, v.y/len, v.z/len );
}

/*
These two functions, MainBulb(..) and IntersectMBulb(..)
do all of the work.  Responsible for the fractal shape/details
*/

vec4 MainBulb(vec4 v)
{
    double r = length( vec3(v.x, v.y, v.z) );
    if (r>BAILOUT) return v;
    double theta = acos(clamp(v.z / r, -1.0, 1.0))*myPower;
    double phi = atan2(v.y, v.x)*myPower;
    v.w = pow(r, myPower - 1.0)*myPower*v.w + 1.0;

    double zr = pow(r, myPower);

    vec3 vv = vec3(sin(theta)*cos(phi), sin(phi)*sin(theta), cos(theta))*zr;
    return vec4(vv.x, vv.y, vv.z, v.w);
}

double IntersectMBulb(vec3& rO, vec3& rD, vec4& trap)
{
    double dist;
    for (int i = 0; i<maxDetailIter; i++) {
        double r = 0.0;
        vec4 v = vec4(rO.x, rO.y, rO.z, 1.0);

        vec3 va (v.x, v.y, v.z);

        trap = vec4(abs(v.x), abs(v.y), abs(v.z), dot(va, va));

        for (int n = 0; n<mmaxIterations; ++n)
        {
            va = vec3(v.x, v.y, v.z);
            r = length(va);
            if (r>BAILOUT) break;

            v = MainBulb(v) + vec4(rO.x, rO.y, rO.z, 0.0);

            va = vec3(v.x, v.y, v.z);
            vec4 v4 (abs(va.x), abs(va.y), abs(va.z), dot(va, va));
            trap.x = min(trap.x, v4.x);
            trap.y = min(trap.y, v4.y);
            trap.z = min(trap.z, v4.z);
            trap.w = min(trap.w, v4.w);
        }

        dist = 0.5*log(r)*r / v.w;
        rO = rO + rD * dist;
        if (dist < EPS) break;
    }
    return dist;
}

/*
  Hacky code to work out an approximate normal for our
  geometry (without lighting it looks horrible)
*/
vec3 NormEstimateMB(vec3 p) {

    double ddd = 0.000001;

    vec4 g0 = vec4(p, .0);
    vec4 gx = vec4(p + vec3(ddd, 0, 0), .0);
    vec4 gy = vec4(p + vec3(0, ddd, 0), .0);
    vec4 gz = vec4(p + vec3(0, 0, ddd), .0);

    vec4 v0 = g0;
    vec4 vx = gx;
    vec4 vy = gy;
    vec4 vz = gz;

    double ln;
    for (int i = 0; i < 100; i++) {
        g0 = MainBulb(g0) + v0;
        gx = MainBulb(gx) + vx;
        gy = MainBulb(gy) + vy;
        gz = MainBulb(gz) + vz;
        ln = length( vec3(g0.x, g0.y, g0.z) );
        if (ln>BAILOUT) break;
    }

    //float ln    = length( vec3(g0.x, g0.y, g0.z) );
    double gradX = length( vec3(gx.x, gx.y, gx.z) ) - ln;
    double gradY = length( vec3(gy.x, gy.y, gy.z) ) - ln;
    double gradZ = length( vec3(gz.x, gz.y, gz.z) ) - ln;
    //N = normalize(vec3(length(gx-g0), length(gy-g0), length(gz-g0)));
    return normalize(vec3(gradX, gradY, gradZ));
}

/*
  Good old Phong lighting function
*/
vec3 Phong(vec3 light, vec3 eye, vec3 pt, vec3 N)
{
    vec3 diffuse = vec3(0.40, 0.95, 0.25); // base color of shading
    const int specularExponent = 10;       // shininess of shading
    const double specularity = 0.45;       // amplitude of specular highlight

    vec3 L = normalize(light - pt);        // find the vector to the light
    vec3 E = normalize(eye - pt);          // find the vector to the eye
    double  NdotL = dot(N, L);             // find the cosine of the angle between light and normal
    vec3 R = L - N * 2 * NdotL;            // find the reflected vector

    diffuse = diffuse + N*0.1;             // add some of the normal for 'effect'

    return diffuse * max(NdotL, 0.0) + specularity*pow(max(dot(E, R), 0), specularExponent);
}

/*
  Program entry point - loop over all the image pixels
  and work out the color (i.e., based on a ray we shoot
  at the Mandelbulb fractal/intersection).
*/
int main(int argc, char *argv[]) {

    int w = 1024, h = 1014; // # default resolution
    if ( argc == 2 ) w = h = atoi(argv[1]); // # resolution

    vec3 *c = new vec3[w*h]; // output colors

#pragma omp parallel for schedule(dynamic, 1) private(r)       // OpenMP
    for (int y=0; y < h; y++ ) {       // Loop over image rows
        fprintf(stderr, "\rRendering (%dx%d res) %5.2f%%", w, h, 100.*y / (h - 1));
        for (unsigned short x=0; x < w; x++ )   // Loop cols
        {
            // index for this pixel into color array c
            const int i = (h - y - 1)*w + x;
            // we 'orientate' the ray so it's not just facing
            // down the z-axis. Instead of implementing all the rotation
            // code, we use some simple vector trigonometry to work out
            // the look at point, direction and origin of the ray
            vec3 off(0,-2,-2);
            vec3 dir = vec3(0,0,0) - off;
            dir = normalize(dir);

            off = dir*-2.5;

            vec3 left = vec3(0,1,0).cross( dir );
            left = normalize(left);

            vec3 up = left.cross( dir );
            up = normalize(up);

            // pixels for the screen -0.5 to 0.5
            double dx = -0.5 + ((x / double(w - 1)) * 1.0);
            double dy = -0.5 + ((y / double(h - 1)) * 1.0);

            vec3 rO = off;
            vec3 rD = dir + left*dx + up*dy;
            rD = rD.norm();

            #if 0 // debug - if we want the camera to look down the z axis
            dx = -0.5 + ((x / double(w - 1) ) * 1.0 );
            dy = -0.5 + ((y / double(h - 1)) * 1.0 );
            rD = vec3(dx, dy, 0.9).norm();
            rO = vec3(0,0,-2.3);
            #endif

            vec4 cols(0,0,0,0);
            // distance from the ray to the geometry
            double dist = IntersectMBulb(rO, rD, cols);

            c[i] = vec3(); // default black
            if ( dist < 0.001 )
            {
                // if we get here, then we have 'hit' the geometry, so we can work
                // out some lighting information.
                // If we just set the color to some 'constant' we won't see any of
                // that sexy detail/geometry that is fractals possess
                vec3 light (2,2,-5); // light position

                vec3 N = NormEstimateMB(rO);

                // Compute the Phong illumination at the point of intersection.
                vec3 color = Phong(light, rD, rO, N);
                c[i] = color;
            }
        }
    }

    // details on the ppm image format
    // ref: https://en.wikipedia.org/wiki/Netpbm#PPM_example

    FILE *f = fopen("image.ppm", "w");         // Write image to PPM file.
    fprintf(f, "P3\n%d %d\n%d\n", w, h, 255);
    for (int i = 0; i < w*h; i++)
        fprintf(f, "%d %d %d ", toInt(c[i].x), toInt(c[i].y), toInt(c[i].z));
}

2D Cross-section Of (3D) Mandelbrot Fractal

MandelBulb Ray Tracer (Stripped)

References and Resources