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 location
float3 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; // discriminant
tmin = tmax = 0.0;
if (disc > 0.0) {
// Real root of disc, so intersection
float sdisc = sqrt(disc);
float t0 = -b - sdisc; // closest intersection distance
float t1 = -b + sdisc; // furthest intersection distance
if (t0 >= 0.0) {
// Ray intersects front of sphere
float min_dist;
float3 z = origin + t0 * direction;
tmin = DE(z, min_dist);
tmax = t0 + t1;
} else if (t0 < 0.0) {
// Ray starts inside sphere
float 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 point
float3 estimate_normal(float3 z, float e)
{
float min_dst; // Not actually used in this particular case
float3 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 contribution
float3 color = float3(0, 0, 0);
specular = 0.0;
float3 L = normalize(light * objRotation - pt); // find the vector to the light
float NdotL = dot(N, L); // find the cosine of the angle between light and normal
if (NdotL > 0.0) {
// Diffuse shading
diffuse = colorDiffuse + abs(N) * colorSpread;
diffuse *= colorLight * NdotL;
// Phong highlight
float3 E = normalize(eye - pt); // find the vector to the eye
float3 R = L - 2.0 * NdotL * N; // find the reflected vector
float 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 point
float2 estimate_texture(float3 zi)
{
float2 uv=float2(0,0);
float3 z=normalize(zi);
float min_dst; // Not actually used in this particular case
float3 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;
else
u = 1.0 - theta;
uv.x=u;
return uv;
}
// Define the ray direction from the pixel coordinates
float3 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 5
i = int( h );
f = h - float(i); // factorial part of h
p = 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 pixel
float4 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 factor
int max_steps = int(float(stepLimit) / epsilonScale);
int i;
float f;
for (i = 0; i < max_steps; ++i) {
dist = DE(ray, min_dist);
// March ray forward
f = epsilonScale * dist;
ray += f * ray_direction;
ray_length += f * dist;
// Are we within the intersection threshold or completely missed the fractal
if (dist < eps || ray_length > tmax) break;
// Set the intersection threshold as a function of the ray length from the camera
eps = 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 forward
f = epsilonScale * dist;
ray += f * light_direction;
// Are we within the intersection threshold or completely missed the fractal
if (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 shadow
color.rgb += specular;
}
} else {
color.rgb += specular;
}
} else {
// Just use the base colour
color.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 pixels
void evaluateDependents()
{
aspectRatio = float(size.x) / float(size.y);
// Camera orientation
float 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 rotation
c1 = 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 sampling
sampleStep = 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 loop
void 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 pixel
for (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。
