我的三维mandelbulb制作fractal成长之路[续]

网络上一般的mandelbulb的代码会比较大，不仅仅画mandelbullb,还画其它，并且增加了许多特效，不适合入门。

目前找到 https://github.com/vpmedia/flash-pbk-collection中MandelbulbQuick.pbk，是入门ray match画mandelbulb比较方便的入门代码。这是基于adobe 的Pixel Bender Toolkit的。

基于它可以实现在mandelbulb上裹一张图片，或者根据位置高度着不同颜色。

效果实现如下：

1.外面裹图片。

2. 根据位置高度着不同颜色

那么基于这个起点，就可以随意做自己的三维分形图了。

修改颜色搭配，power>8的情况。

修改后的pbk代码：

/*** MandelbulbQuick.pbk* Last update: 14 December 2009** Changelog:*        1.0     - Initial release*      1.0.1   - Fixed a missing asymmetry thanks to Chris King (http://www.dhushara.com)*             - Refinements in the colouring*      1.0.2   - Added radiolaria option for a funky hair-like effect*                - Incorporated the scalar derivative method as described here:*             - http://www.fractalforums.com/mandelbulb-implementation/realtime-renderingoptimisations/*      1.0.3   - Created a quick version of the script as using a boolean flag to determine*                 which distance estimation method created long compilation times.*         1.0.4   - Fixed issue with older graphic cards and the specular highlights** * Copyright (c) 2009 Tom Beddard* http://www.subblue.com** For more Flash and PixelBender based generative graphics experiments see:* http://www.subblue.com/blog** Licensed under the MIT License:* http://www.opensource.org/licenses/mit-license.php*** Credits and references* ======================* For the story behind the 3D Mandelbrot see the following page:* http://www.skytopia.com/project/fractal/mandelbulb.html** The original forum disussion with many implementation details can be found here:* http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/** This implementation references the 4D Quaternion GPU Raytracer by Keenan Crane:* http://www.devmaster.net/forums/showthread.php?t=4448** and the NVIDIA CUDA/OptiX implementation by cbuchner1:* http://forums.nvidia.com/index.php?showtopic=150985**/#define PI 3.141592653
#define MIN_EPSILON 3e-7<languageVersion : 1.0;>kernel MandelbulbQuick
<    namespace : "com.subblue.filters";vendor : "Tom Beddard";version : 1;description : "Mandelbulb Fractal Ray Tracer - the quick version";
>{parameter int antialiasing<minValue:1;maxValue:3;defaultValue:1;description:"Super sampling quality. Number of samples squared per pixel.";>;parameter bool phong<defaultValue:true;description: "Enable phong shading.";>;parameter bool julia<defaultValue:false;description: "Enable Julia set version.";>;parameter bool radiolaria<defaultValue:false;description: "Enable radiolaria style.";>;parameter float radiolariaFactor<minValue: -4.0;maxValue: 4.0;defaultValue: 0.0;description: "Tweak the radiolaria effect.";>;parameter float shadows<minValue:0.0;maxValue:1.0;defaultValue:0.0;description: "Enable ray traced shadows.";>;parameter float ambientOcclusion<minValue:0.0;maxValue:1.0;defaultValue:0.8;description: "Enable fake ambient occlusion factor based on the orbit trap.";>;parameter float ambientOcclusionEmphasis<minValue:0.0;maxValue:1.0;defaultValue:0.58;description: "Emphasise the structure edges based on the number of steps it takes to reach a point in the fractal.";>;parameter float bounding<minValue:1.0;maxValue:16.0;defaultValue:2.5;description: "Sets the bounding sphere radius to help accelerate the raytracing.";>;parameter float bailout<minValue:0.5;maxValue:12.0;defaultValue:4.0;description: "Sets the bailout value for the fractal calculation. Lower values give smoother less detailed results.";>;parameter float power<minValue:2.0;maxValue:32.0;defaultValue:8.0;description: "The power of the fractal.";>;parameter float textureRate<minValue:0.0;maxValue:1.0;defaultValue:0.210;description: "rate of texture.";>; parameter float textureStartc<minValue:0.0;maxValue:380.0;defaultValue:10.50;description: "start H of texture.";>;      parameter float2 textureSize<minValue: float2(50.0, 50.0);maxValue:float2(2500.0, 1500.0);defaultValue:float2(500.0, 250.0);description: "Tweak the mapping of the triplex numbers into spherical co-ordinates - in other words tweak the surface shape.";>;parameter float3 julia_c<minValue:float3(-2, -2, -2);maxValue:float3(2, 2, 2);defaultValue:float3(1.0, 0.0, 0.0);description: "The c constant for Julia set fractals";>;parameter float3 cameraPosition<minValue:float3(-4, -4, -4);maxValue:float3(4, 4, 4);defaultValue:float3(0, -2.6, 0);description: "Camera position.";>;parameter float3 cameraPositionFine<minValue:float3(-0.1, -0.1, -0.1);maxValue:float3(0.1, 0.1, 0.1);defaultValue:float3(0, 0.0, 0.0);description: "Fine tune position.";>;parameter float3 cameraRotation<minValue:float3(-180, -180, -180);maxValue:float3(180, 180, 180);defaultValue:float3(0, 0, -90);description: "Pointing angle in each axis of the camera.";>;parameter float cameraZoom<minValue:0.0;maxValue:10.0;defaultValue:0.0;description: "Zoom the camera view.";>;parameter float3 light<minValue:float3(-50, -50, -50);maxValue:float3(50, 50, 50);defaultValue:float3(38, -42, 38);description: "Position of point light.";>;parameter float3 colorBackground<minValue:float3(0, 0, 0);maxValue:float3(1, 1, 1);defaultValue:float3(0.0, 0.0, 0.0);description: "Background colour.";aeUIControl: "aeColor";>;parameter float colorBackgroundTransparency<minValue:float(0.0);maxValue:float(1.0);defaultValue:float(1.0);description: "Background transparency.";>;parameter float3 colorDiffuse<minValue:float3(0, 0, 0);maxValue:float3(1, 1, 1);defaultValue:float3(0.0, 0.85, 0.99);description: "Diffuse colour.";aeUIControl: "aeColor";>;parameter float3 colorAmbient<minValue:float3(0, 0, 0);maxValue:float3(1, 1, 1);defaultValue:float3(0.67, 0.85, 1.0);description: "Ambient light colour.";aeUIControl: "aeColor";>;parameter float colorAmbientIntensity<minValue:float(0);maxValue:float(1);defaultValue:float(0.4);description: "Ambient light intensity.";>;parameter float3 colorLight<minValue:float3(0, 0, 0);maxValue:float3(1, 1, 1);defaultValue:float3(0.48, 0.59, 0.66);description: "Light colour.";aeUIControl: "aeColor";>;parameter float colorSpread<minValue:0.0;maxValue:1.0;defaultValue:0.2;description: "Vary the colour based on the normal direction.";>;parameter float rimLight<minValue:0.0;maxValue:1.0;defaultValue:0.0;description: "Rim light factor.";>;parameter float specularity<minValue:0.0;maxValue:1.0;defaultValue:0.66;description: "Phone specularity";>;parameter float specularExponent<minValue:0.1;maxValue:50.0;defaultValue:15.0;description: "Phong shininess";>;parameter float3 rotation<minValue:float3(-180, -180, -180);maxValue:float3(180, 180, 180);defaultValue:float3(0, 36, 39.6);description: "Rotate the Mandelbulb in each axis.";>;parameter int maxIterations<minValue:1;maxValue:20;defaultValue:6;description: "More iterations reveal more detail in the fractal surface but takes longer to calculate.";>;parameter int stepLimit<minValue:10;maxValue:200;defaultValue:110;description: "The maximum number of steps a ray should take.";>;parameter float epsilonScale<minValue:0.1;maxValue:1.0;defaultValue:1.0;description: "Scale the epsilon step distance. Smaller values are slower but will generate smoother results for thin areas.";>;parameter int2 size<minValue:int2(100, 100);maxValue:int2(2048, 2048);defaultValue:int2(640, 480);description: "The output size in pixels.";>;region generated(){return region(float4(0, 0, size.x, size.y));}dependent float bailout_sr, aspectRatio, sampleStep, sampleContribution, pixel_scale, eps_start;dependent float3 eye;dependent float3x3 viewRotation, objRotation;input image4 src;output pixel4 dst;// Scalar derivative approach by Enforcer:// http://www.fractalforums.com/mandelbulb-implementation/realtime-renderingoptimisations/void powN(inout float3 z, float zr0, inout float dr){float zo0 = asin(z.z / zr0);float zi0 = atan(z.y, z.x);float zr = pow(zr0, power - 1.0);float zo = (zo0) * power;float zi = (zi0) * power;float czo = cos(zo);dr = zr * dr * power + 1.0;zr *= zr0;z = zr * float3(czo*cos(zi), czo*sin(zi), sin(zo));}// The fractal calculation//// Calculate the closest distance to the fractal boundary and use this// distance as the size of the step to take in the ray marching.//// Fractal formula://      z' = z^p + c//// For each iteration we also calculate the derivative so we can estimate// the distance to the nearest point in the fractal set, which then sets the// maxiumum step we can move the ray forward before having to repeat the calculation.////    dz' = p * z^(p-1)//// The distance estimation is then calculated with:////   0.5 * |z| * log(|z|) / |dz|//float DE(float3 z0, inout float min_dist){float3 c = julia ? julia_c : z0; // Julia set has fixed c, Mandelbrot c changes with locationfloat3 z = z0;float dr = 1.0;float r  = length(z);if (r < min_dist) min_dist = r;for (int n = 0; n < maxIterations; n++) {powN(z, r, dr);z += c;if (radiolaria && z.y > radiolariaFactor) z.y = radiolariaFactor;r = length(z);if (r < min_dist) min_dist = r;if (r > bailout) break;}return 0.5 * log(r) * r / dr;}// Intersect bounding sphere//// If we intersect then set the tmin and tmax values to set the start and// end distances the ray should traverse.bool intersectBoundingSphere(float3 origin,float3 direction,out float tmin,out float tmax){bool hit = false;float b = dot(origin, direction);float c = dot(origin, origin) - bounding;float disc = b*b - c;         // discriminanttmin = tmax = 0.0;if (disc > 0.0) {// Real root of disc, so intersectionfloat sdisc = sqrt(disc);float t0 = -b - sdisc;           // closest intersection distancefloat t1 = -b + sdisc;            // furthest intersection distanceif (t0 >= 0.0) {// Ray intersects front of spherefloat min_dist;float3 z = origin + t0 * direction;tmin = DE(z, min_dist);tmax = t0 + t1;} else if (t0 < 0.0) {// Ray starts inside spherefloat min_dist;float3 z = origin;tmin = DE(z, min_dist);tmax = t1;}hit = true;}return hit;}// Calculate the gradient in each dimension from the intersection pointfloat3 estimate_normal(float3 z, float e){float min_dst;   // Not actually used in this particular casefloat3 z1 = z + float3(e, 0, 0);float3 z2 = z - float3(e, 0, 0);float3 z3 = z + float3(0, e, 0);float3 z4 = z - float3(0, e, 0);float3 z5 = z + float3(0, 0, e);float3 z6 = z - float3(0, 0, e);float dx = DE(z1, min_dst) - DE(z2, min_dst);float dy = DE(z3, min_dst) - DE(z4, min_dst);float dz = DE(z5, min_dst) - DE(z6, min_dst);return normalize(float3(dx, dy, dz) / (2.0*e));}// Computes the direct illumination for point pt with normal N due to// a point light at light and a viewer at eye.float3 Phong(float3 pt, float3 N, out float specular){float3 diffuse  = float3(0, 0, 0);         // Diffuse contributionfloat3 color = float3(0, 0, 0);specular = 0.0;float3 L = normalize(light * objRotation - pt); // find the vector to the lightfloat  NdotL = dot(N, L);           // find the cosine of the angle between light and normalif (NdotL > 0.0) {// Diffuse shadingdiffuse = colorDiffuse + abs(N) * colorSpread;diffuse *= colorLight * NdotL;// Phong highlightfloat3 E = normalize(eye - pt);        // find the vector to the eyefloat3 R = L - 2.0 * NdotL * N;       // find the reflected vectorfloat  RdE = dot(R,E);if (RdE <= 0.0) {specular = specularity * pow(abs(RdE), specularExponent);}} else {diffuse = colorDiffuse * abs(NdotL) * rimLight;}return (colorAmbient * colorAmbientIntensity) + diffuse;}// Calculate the gradient in each dimension from the intersection pointfloat2 estimate_texture(float3 zi){float2 uv=float2(0,0);float3 z=normalize(zi);float min_dst;   // Not actually used in this particular casefloat3 z1 =float3(0, 0, 1.0);float3 y1 =float3(1.0, 0.0, 0);float phi = acos( dot( z, z1 ));uv.y = phi / PI;float u=0.0;if(abs(phi)<0.05)phi=0.05;float theta = ( acos( dot( z, y1 ) / sin( phi )) ) / ( 2.0 * PI) ;if( dot( cross( z1, y1 ), z )>0.0 )u = theta;elseu = 1.0 - theta;uv.x=u;return uv;}// Define the ray direction from the pixel coordinatesfloat3 rayDirection(float2 p){float3 direction = float3( 2.0 * aspectRatio * p.x / float(size.x) - aspectRatio,-2.0 * p.y / float(size.y) + 1.0,-2.0 * exp(cameraZoom));return normalize(direction * viewRotation * objRotation);}//h=(0.0,360.0) s=(0.0,1.0) v=(0.0,1.0)float4 HSVToRGB(float h,float s,float v){float4 color=float4(0,0,0,1);float angle;const float f25=255.0;int i;int clra,clrb,clrc;h=mod(h,360.0)/60.0;i=int(h);angle=h-float(i);clra=int(v*(1.0 - s)*f25);clrb=int(v*(1.0 - s*angle)*f25);clrc=int(v*(1.0 - s*(1.0-angle))*f25);if(i==0){color.rgb=float3(v*f25,clrc,clra);}else if(i==1){color.rgb=float3(clrb,v*f25,clra);}  else if(i==2){color.rgb=float3(clra,v*f25,clrc);}  else if(i==3){color.rgb=float3(clra,clrb,v*f25);}   else if(i==4){color.rgb=float3(clrc,clra,v*f25);} else {color.rgb=float3(v*f25,clra,clrb);}return color;}void HSVtoRGB1(out float  r, out float   g, out float  b, float h, float s, float v ){int i;float f25=1.0;//255.0;float f, p, q, t;h=mod(h,360.0);if( s == 0.0 ) {// achromatic (grey)r =  g =  b = v;} else {h /= 60.0;            // sector 0 to 5i = int( h );f = h - float(i);          // factorial part of hp = v * ( 1.0 - s );q = v * ( 1.0 - s * f );t = v * ( 1.0  - s * ( 1.0 - f ) );if(i==0){r = v;g = t;b = p;}else if(i==1){r = q;g = v;b = p;}else if(i==2){r = p;g = v;b = t;}else if(i==3){r = p;g = q;b = v;}else if(i==4){r = t;g = p;b = v;}else {r = v;g = p;b = q;}}r*=f25;g*=f25;b*=f25;}float4  HSVtoRGB3(float h, float s, float v) {float4 c =float4(0,0,0,1.0);HSVtoRGB1(c.r,c.g,c.b,h,s,v);return c;}// Calculate the output colour for each input pixelfloat4 renderPixel(float2 pixel,inout float4  textureColor){float tmin, tmax;float3 ray_direction = rayDirection(pixel);float4 color;color.rgb = colorBackground.rgb;color.a = colorBackgroundTransparency;if (intersectBoundingSphere(eye, ray_direction, tmin, tmax)) {float3 ray = eye + tmin * ray_direction;float dist, ao;float min_dist = 4.0;float ray_length = tmin;float eps = MIN_EPSILON;// number of raymarching steps scales inversely with factorint max_steps = int(float(stepLimit) / epsilonScale);int i;float f;for (i = 0; i < max_steps; ++i) {dist = DE(ray, min_dist);// March ray forwardf = epsilonScale * dist;ray += f * ray_direction;ray_length += f * dist;// Are we within the intersection threshold or completely missed the fractalif (dist < eps || ray_length > tmax) break;// Set the intersection threshold as a function of the ray length from the cameraeps = max(MIN_EPSILON, pixel_scale * ray_length);}// Found intersection?if (dist < eps) {ao  = 1.0 - clamp(1.0 - min_dist * min_dist, 0.0, 1.0) * ambientOcclusion;if (phong) {float3 normal = estimate_normal(ray, eps/2.0);float specular = 0.0;float2 uvdst=float2(0,0);float2 uv=estimate_texture(ray);float2 srcsize=textureSize;//pixelSize(src);uvdst.x=uv.x*srcsize.x;uvdst.y=uv.y*srcsize.y;//uvdst=mod(uvdst,srcsize);//textureColor=sampleNearest(src, uvdst); color.rgb = Phong(ray, normal, specular).rgb;if (shadows > 0.0) {// The shadow ray will start at the intersection point and go// towards the point light. We initially move the ray origin// a little bit along this direction so that we don't mistakenly// find an intersection with the same point again.float3 light_direction = normalize((light - ray) * objRotation);ray += normal * eps * 2.0;float min_dist2;dist = 4.0;for (int j = 0; j < max_steps; ++j) {dist = DE(ray, min_dist2);// March ray forwardf = epsilonScale * dist;ray += f * light_direction;// Are we within the intersection threshold or completely missed the fractalif (dist < eps || dot(ray, ray) > bounding * bounding) break;}// Again, if our estimate of the distance to the set is small, we say// that there was a hit and so the source point must be in shadow.if (dist < eps) {color.rgb *= 1.0 - shadows;} else {// Only add specular component when there is no shadowcolor.rgb += specular;}} else {color.rgb += specular;}} else {// Just use the base colourcolor.rgb = colorDiffuse;}float rr=length(ray);//textureColor.rgb=HSVtoRGB3(textureStartc+ mod(rr*100.0,30.0)*8.0,0.5,0.5).rgb;    //float3(40,0,140)*0.5; // textureColor.rgb=HSVtoRGB3(textureStartc+ rr*150.0,0.9,0.9).rgb;   //textureColor.rgb=HSVtoRGB3(textureStartc+ rr*150.0,rr,0.9).rgb;                 //ao *= 1.0 - (float(i) / float(max_steps)) * ambientOcclusionEmphasis * 2.0;color.rgb *= ao;color.a = 1.0;}}return clamp(color, 0.0, 1.0);}// Common values used by all pixelsvoid evaluateDependents(){aspectRatio = float(size.x) / float(size.y);// Camera orientationfloat c1 = cos(radians(-cameraRotation.x));float s1 = sin(radians(-cameraRotation.x));float3x3 viewRotationY = float3x3( c1, 0, s1,0, 1, 0,-s1, 0, c1);float c2 = cos(radians(-cameraRotation.y));float s2 = sin(radians(-cameraRotation.y));float3x3 viewRotationZ = float3x3( c2, -s2, 0,s2, c2, 0,0, 0, 1);float c3 = cos(radians(-cameraRotation.z));float s3 = sin(radians(-cameraRotation.z));float3x3 viewRotationX = float3x3( 1, 0, 0,0, c3, -s3,0, s3, c3);viewRotation = viewRotationX * viewRotationY * viewRotationZ;// Object rotationc1 = cos(radians(-rotation.x));s1 = sin(radians(-rotation.x));float3x3 objRotationY = float3x3( c1, 0, s1,0, 1, 0,-s1, 0, c1);c2 = cos(radians(-rotation.y));s2 = sin(radians(-rotation.y));float3x3 objRotationZ = float3x3( c2, -s2, 0,s2, c2, 0,0, 0, 1);c3 = cos(radians(-rotation.z));s3 = sin(radians(-rotation.z));float3x3 objRotationX = float3x3( 1, 0, 0,0, c3, -s3,0, s3, c3);objRotation = objRotationX * objRotationY * objRotationZ;eye = (cameraPosition + cameraPositionFine);if (eye == float3(0, 0, 0)) eye = float3(0, 0.0001, 0);eye *= objRotation;// Super samplingsampleStep = 1.0 / float(antialiasing);sampleContribution = 1.0 / pow(float(antialiasing), 2.0);pixel_scale = 1.0 / max(float(size.x), float(size.y));}// The main loopvoid evaluatePixel(){float4 color = float4(0, 0, 0, 0);float frate=0.001;float4  textureColor= float4(0, 0, 0, 0);if (antialiasing > 1) {// Average antialiasing^2 points per pixelfor (float i = 0.0; i < 1.0; i += sampleStep)for (float j = 0.0; j < 1.0; j += sampleStep)color += sampleContribution * renderPixel(float2(outCoord().x + i, outCoord().y + j),textureColor);} else {color = renderPixel(outCoord(),textureColor);}// Return the final color which is still the background color if we didn't hit anything.if(dot(color.rgb,color.rgb)>0.0){ frate=textureRate; //*0.01;color.rgb=color.rgb*(1.0-frate)+textureColor.rgb*frate;}dst = color;}
}

分形背后的数学公式推导:见下面整理文档。

https://download.csdn.net/download/foolpanda1168/10913752

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