数据可视化1—蒙特卡罗光线追踪

蒙特卡罗光线追踪算法 smallpt: Global Illumination in 99 lines of C++

1.光的基本传递模型

（1）在一个要渲染的场景中，我们认为光能由预先指定的光源发出，然后我们以光线来描述光能的传递过程，当整个场景中的光能信息被我们计算出来后，我们收集这些信息转化为顶点的亮度。　　

（2）光线经过物体表面可以产生反射和漫反射，光线透过物体可以产生折射和散射。具体产生哪种出射效果，依据物体的表面属性而定。物体的表面一般不会是理想的某种单一属性的表面，表面可以同时存在反射，折射，漫反射等多种属性，各种属性按一定比例混合之后才是其表面反射模型。　　

（3）一点的在某一个视线方向上的光亮度＝该点在该方向的自身发光亮度＋半球入射光能在该方向所产生的反射光亮度. 　

（4）关于散射，高度真实的散射是一个很难模拟的物理过程，一般在渲染中都不会采用过于复杂的物理模型来表示散射，而是采用一些取巧的办法来计算散射。　　

（5）在常见的渲染中，有两种效果很难模拟，但是它们会使人眼觉得场景更真实。　　

[1]color bleeding :入射光为漫反射，受光表面属性为漫反射，出射光是漫反射。比如把一本蓝色的纸制的书靠近白色的墙，墙上会有浅浅的蓝晕。　　

[2]caustics:入射光为镜面反射或折射，受光表面属性为漫反射，出射光是漫反射。比如把一个装了红色葡萄酒的酒杯放在木桌上面，会有光透过杯中的酒在桌上形成一块很亮的红色区域。

2.蒙特卡罗光线追踪

（1）蒙特卡罗光线追踪对逆向光线追踪模型进行改进，其中最大的区别在于把概率模型引入光线追踪。逆向光线追踪中物体的表面材质很单一。引入俄罗斯赌盘轮，可以设定漫反射、镜面反射、甚至折射的概率。丰富表面材质的显示每个像素点只采样一条光线计算出的颜色，正确率不高。引入蒙特卡罗可以多次采样求平均，优化渲染结果。

（2）对传统的逆向光线追踪的改进　　

传统的逆向光线追踪算法有两个突出的缺点，就是表面属性的单一，和不考虑漫反射。我们不难通过模型的修正来缓解这两个问题。我们首先认为一个表面的属性可以是混合的，比如它有20%的成分是反射，30%的成分是折射，50%的成分是漫反射。这里的百分比可以这样理解，当一根光线打在该表面后，它有20%的概率发生反射，30%的概率发生折射，50%的概率发生漫反射。然后我们通过多次计算光线跟踪，每次按照概率决定光线的反射属性，这样在就把漫反射也考虑了进去。

具体的算法如下：　　

①从视点出发，经过投影屏幕上的每一个像素向场景发射一根虚拟的光线。　　

②当光线与景物相交时按照俄罗斯轮盘赌规则决定他的反射属性。　　

③根据不同的反射属性继续跟踪计算，直到正常结束或者异常结束。如果反射的属性为漫反射，则随机选择一个反射方向进行跟踪。　　

④重复前面的过程，把每次渲染出来的贴图逐像素叠加混合，直到渲染出的结果达到满意程度。

该方法是一种比较简易的基于物理模型的渲染，其本质就是通过大量的随机采样来模拟半球积分。这种方法在光照细节上可以产生真实度很高的图像，但是图像质量有比较严重的走样，而且效率极其低下。

3.99行C++代码实现蒙特卡罗光线追踪算法

#include <math.h>   // smallpt, a Path Tracer by Kevin Beason, 2008
#include <stdlib.h> // Make : g++ -O3 -fopenmp smallpt.cpp -o smallpt
#include <stdio.h>  //        Remove "-fopenmp" for g++ version < 4.2 #ifndef _USE_MATH_DEFINES
#define _USE_MATH_DEFINES
#include <math.h>
#else
#include <cmath>
#endif // _USE_MATH_DEFINESdouble erand48 (unsigned short xsubi[3]) {return rand() / (double)RAND_MAX;
}struct Vec {        // Usage: time ./smallpt 5000 && xv image.ppm double x, y, z;                  // position, also color (r,g,b) Vec(double x_=0, double y_=0, double z_=0){ x=x_; y=y_; z=z_; } Vec operator+(const Vec &b) const { return Vec(x+b.x,y+b.y,z+b.z); } Vec operator-(const Vec &b) const { return Vec(x-b.x,y-b.y,z-b.z); } Vec operator*(double b) const { return Vec(x*b,y*b,z*b); } Vec mult(const Vec &b) const { return Vec(x*b.x,y*b.y,z*b.z); } Vec& norm(){ return *this = *this * (1/sqrt(x*x+y*y+z*z)); } double dot(const Vec &b) const { return x*b.x+y*b.y+z*b.z; } // cross: Vec operator%(Vec&b){return Vec(y*b.z-z*b.y,z*b.x-x*b.z,x*b.y-y*b.x);}
}; //ray 射线
struct Ray { Vec o, d; Ray(Vec o_, Vec d_) : o(o_), d(d_) {} };
enum Refl_t { DIFF, SPEC, REFR };  // material types, used in radiance()
struct Sphere { double rad;       // radius Vec p, e, c;      // position, emission, color Refl_t refl;      // reflection type (DIFFuse, SPECular, REFRactive) Sphere(double rad_, Vec p_, Vec e_, Vec c_, Refl_t refl_): rad(rad_), p(p_), e(e_), c(c_), refl(refl_) {} double intersect(const Ray &r) const { // returns distance, 0 if nohit Vec op = p-r.o; // Solve t^2*d.d + 2*t*(o-p).d + (o-p).(o-p)-R^2 = 0 double t, eps=1e-4, b=op.dot(r.d), det=b*b-op.dot(op)+rad*rad; if (det<0) return 0; else det=sqrt(det); return (t=b-det)>eps ? t : ((t=b+det)>eps ? t : 0); }
}; //球体 diff漫反射  spec镜面反射  refr反射和折射
Sphere spheres[] = {//Scene: radius, position, emission, color, material Sphere(1e5, Vec( 1e5+1,40.8,81.6), Vec(),Vec(.75,.25,.25),DIFF),//Left Sphere(1e5, Vec(-1e5+99,40.8,81.6),Vec(),Vec(.25,.25,.75),DIFF),//Rght Sphere(1e5, Vec(50,40.8, 1e5),     Vec(),Vec(.75,.75,.75),DIFF),//Back Sphere(1e5, Vec(50,40.8,-1e5+170), Vec(),Vec(1,1,1)*.999,DIFF),//Frnt Sphere(1e5, Vec(50, 1e5, 81.6),    Vec(),Vec(.75,.75,.75),DIFF),//Botm Sphere(1e5, Vec(50,-1e5+81.6,81.6),Vec(),Vec(.75,.75,.75),DIFF),//Top Sphere(16.5,Vec(27,16.5,47),       Vec(),Vec(1,1,1)*.999, SPEC),//Mirr Sphere(16.5,Vec(73,16.5,78),       Vec(),Vec(1,1,1)*.999, REFR),//GlasSphere(10,Vec(40,50,100),       Vec(),Vec(.75,.75,.75), DIFF),//DIFFSphere(600, Vec(50,681.6-.27,81.6),Vec(12,12,12),  Vec(), DIFF) //Lite
}; //遍历所有的球，求交点
inline double clamp(double x){ return x<0 ? 0 : x>1 ? 1 : x; }
inline int toInt(double x){ return int(pow(clamp(x),1/2.2)*255+.5); }
inline bool intersect(const Ray &r, double &t, int &id){ double n=sizeof(spheres)/sizeof(Sphere), d, inf=t=1e20; for(int i=int(n);i--;) if((d=spheres[i].intersect(r))&&d<t){t=d;id=i;} return t<inf;
} //光线跟踪递归函数
Vec radiance(const Ray &r, int depth, unsigned short *Xi){ double t;                               // distance to intersection int id=0;                               // id of intersected object if (!intersect(r, t, id)) return Vec(); // if miss, return black const Sphere &obj = spheres[id];        // the hit object Vec x=r.o+r.d*t, n=(x-obj.p).norm(), nl=n.dot(r.d)<0?n:n*-1, f=obj.c; double p = f.x>f.y && f.x>f.z ? f.x : f.y>f.z ? f.y : f.z; // max refl if (++depth>5) if (erand48(Xi)<p) f=f*(1/p); else return obj.e; //R.R. if (depth > 100) return Vec();if (obj.refl == DIFF){                  // Ideal DIFFUSE reflection double r1=2*M_PI*erand48(Xi), r2=erand48(Xi), r2s=sqrt(r2); Vec w=nl, u=((fabs(w.x)>.1?Vec(0,1):Vec(1))%w).norm(), v=w%u; Vec d = (u*cos(r1)*r2s + v*sin(r1)*r2s + w*sqrt(1-r2)).norm(); return obj.e + f.mult(radiance(Ray(x,d),depth,Xi)); } else if (obj.refl == SPEC)            // Ideal SPECULAR reflection return obj.e + f.mult(radiance(Ray(x,r.d-n*2*n.dot(r.d)),depth,Xi)); Ray reflRay(x, r.d-n*2*n.dot(r.d));     // Ideal dielectric REFRACTION bool into = n.dot(nl)>0;                // Ray from outside going in? double nc=1, nt=1.5, nnt=into?nc/nt:nt/nc, ddn=r.d.dot(nl), cos2t; if ((cos2t=1-nnt*nnt*(1-ddn*ddn))<0)    // Total internal reflection return obj.e + f.mult(radiance(reflRay,depth,Xi)); Vec tdir = (r.d*nnt - n*((into?1:-1)*(ddn*nnt+sqrt(cos2t)))).norm(); double a=nt-nc, b=nt+nc, R0=a*a/(b*b), c = 1-(into?-ddn:tdir.dot(n)); double Re=R0+(1-R0)*c*c*c*c*c,Tr=1-Re,P=.25+.5*Re,RP=Re/P,TP=Tr/(1-P); return obj.e + f.mult(depth>2 ? (erand48(Xi)<P ?   // Russian roulette radiance(reflRay,depth,Xi)*RP:radiance(Ray(x,tdir),depth,Xi)*TP) : radiance(reflRay,depth,Xi)*Re+radiance(Ray(x,tdir),depth,Xi)*Tr);
} //camera位置（50，52，295.6），往z轴负方向看
int main(int argc, char *argv[]){ int w=1024, h=768, samps=50; // # samples采样 Ray cam(Vec(50,52,295.6), Vec(0,-0.042612,-1).norm()); // cam pos, dir Vec cx=Vec(w*.5135/h), cy=(cx%cam.d).norm()*.5135, r, *c=new Vec[w*h]; #pragma omp parallel for schedule(dynamic, 1) private(r)       // OpenMP //遍历每个像素点，用随机采样的方式求得要射出的光线的方向dfor (int y=0; y<h; y++){                       // Loop over image rows fprintf(stderr,"\rRendering (%d spp) %5.2f%%",samps*4,100.*y/(h-1)); for (unsigned short x=0, Xi[3]={0,0,y*y*y}; x<w; x++)   // Loop cols for (int sy=0, i=(h-y-1)*w+x; sy<2; sy++)     // 2x2 subpixel rows for (int sx=0; sx<2; sx++, r=Vec()){        // 2x2 subpixel cols for (int s=0; s<samps; s++){ double r1=2*erand48(Xi), dx=r1<1 ? sqrt(r1)-1: 1-sqrt(2-r1); double r2=2*erand48(Xi), dy=r2<1 ? sqrt(r2)-1: 1-sqrt(2-r2); Vec d = cx*( ( (sx+.5 + dx)/2 + x)/w - .5) + cy*( ( (sy+.5 + dy)/2 + y)/h - .5) + cam.d; r = r + radiance(Ray(cam.o+d*140,d.norm()),0,Xi)*(1./samps); } // Camera rays are pushed ^^^^^ forward to start in interior c[i] = c[i] + Vec(clamp(r.x),clamp(r.y),clamp(r.z))*.25; }  } 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)); }

4.实验结果图

这是main函数中samps值设定为100时得到的图片，samps值越大清晰度越高，同时运行时间也随之增加，可自行设置samps的值进行测试。

在场景中添加一个塑料球

5.其他场景设计

从左到右从上到下依次为forest——vista——sky——wada

代码：

下面个各个场景的代码实现，可自行将下述某个场景的代码带入上述算法，实现场景验证。

======================================================================
sky
======================================================================
// Idea stolen from Picogen http://picogen.org/ by phresnel/greenhybrid
Vec Cen(50,40.8,-860);
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0Sphere(1600, Vec(1,0,2)*3000, Vec(1,.9,.8)*1.2e1*1.56*2,Vec(), DIFF), // sunSphere(1560, Vec(1,0,2)*3500,Vec(1,.5,.05)*4.8e1*1.56*2, Vec(),  DIFF), // horizon sun2
//   Sphere(10000,Cen+Vec(0,0,-200), Vec(0.0627, 0.188, 0.569)*6e-2*8, Vec(.7,.7,1)*.25,  DIFF), // skySphere(10000,Cen+Vec(0,0,-200), Vec(0.00063842, 0.02001478, 0.28923243)*6e-2*8, Vec(.7,.7,1)*.25,  DIFF), // skySphere(100000, Vec(50, -100000, 0),  Vec(),Vec(.3,.3,.3),DIFF), // grndSphere(110000, Vec(50, -110048.5, 0),  Vec(.9,.5,.05)*4,Vec(),DIFF),// horizon brightenerSphere(4e4, Vec(50, -4e4-30, -3000),  Vec(),Vec(.2,.2,.2),DIFF),// mountains
//  Sphere(3.99e4, Vec(50, -3.99e4+20.045, -3000),  Vec(),Vec(.7,.7,.7),DIFF),// mountains snowSphere(26.5,Vec(22,26.5,42),   Vec(),Vec(1,1,1)*.596, SPEC), // white MirrSphere(13,Vec(75,13,82),   Vec(),Vec(.96,.96,.96)*.96, REFR),// GlasSphere(22,Vec(87,22,24),   Vec(),Vec(.6,.6,.6)*.696, REFR)    // Glas2
};======================================================================
nightsky
======================================================================
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0//     rad       pos                   emis           col     refl
//  Sphere(1e3,   Vec(1,1,-2)*1e4,    Vec(1,1,1)*5e2,     Vec(), DIFF), // moon
//  Sphere(3e2,   Vec(.6,.2,-2)*1e4,    Vec(1,1,1)*5e3,     Vec(), DIFF), //
//  moonSphere(2.5e3,   Vec(.82,.92,-2)*1e4,    Vec(1,1,1)*.8e2,     Vec(), DIFF), // moon//  Sphere(2.5e4, Vec(50, 0, 0),     Vec(1,1,1)*1e-3,    Vec(.2,.2,1)*0.0075, DIFF), // sky
//  Sphere(2.5e4, Vec(50, 0, 0),  Vec(0.114, 0.133, 0.212)*1e-2,  Vec(.216,.384,1)*0.0007, DIFF), // skySphere(2.5e4, Vec(50, 0, 0),  Vec(0.114, 0.133, 0.212)*1e-2,  Vec(.216,.384,1)*0.003, DIFF), // skySphere(5e0,   Vec(-.2,0.16,-1)*1e4, Vec(1.00, 0.843, 0.698)*1e2,   Vec(), DIFF),  // starSphere(5e0,   Vec(0,  0.18,-1)*1e4, Vec(1.00, 0.851, 0.710)*1e2,  Vec(), DIFF),  // starSphere(5e0,   Vec(.3, 0.15,-1)*1e4, Vec(0.671, 0.780, 1.00)*1e2,   Vec(), DIFF),  // starSphere(3.5e4,   Vec(600,-3.5e4+1, 300), Vec(),   Vec(.6,.8,1)*.01,  REFR),   //poolSphere(5e4,   Vec(-500,-5e4+0, 0),   Vec(),      Vec(1,1,1)*.35,  DIFF),    //hillSphere(16.5,  Vec(27,0,47),         Vec(),              Vec(1,1,1)*.33, DIFF), //hutSphere(7,     Vec(27+8*sqrt(2),0,47+8*sqrt(2)),Vec(),  Vec(1,1,1)*.33,  DIFF), //doorSphere(500,   Vec(-1e3,-300,-3e3), Vec(),  Vec(1,1,1)*.351,    DIFF),  //mntSphere(830,   Vec(0,   -500,-3e3), Vec(),  Vec(1,1,1)*.354,    DIFF),  //mntSphere(490,  Vec(1e3,  -300,-3e3), Vec(),  Vec(1,1,1)*.352,    DIFF),  //mnt
};======================================================================
island
======================================================================
// Inspired by cover of "Time Planet Earth: An Illustrated History"
Vec Cen(50,-20,-860);
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0//     rad       pos                   emis           col     reflSphere(160,  Cen+Vec(0, 600, -500),Vec(1,1,1)*2e2, Vec(),  DIFF), // sunSphere(800, Cen+Vec(0,-880,-9120),Vec(1,1,1)*2e1, Vec(),  DIFF), // horizonSphere(10000,Cen+Vec(0,0,-200), Vec(0.0627, 0.188, 0.569)*1e0, Vec(1,1,1)*.4,  DIFF), // sky//  Sphere(1000, Cen+Vec(0,-1080,-8020),Vec(1,1,1)*2e1, Vec(),  DIFF), // horizon
//  Sphere(10000,Cen+Vec(0,0,-200), Vec(0.0627, 0.188, 0.569)*1e0, Vec(1,1,1)*.3,  DIFF), // sky//  Sphere(800, Cen+Vec(0,-720,-200),Vec(),  Vec(0, 0.588, 0.8),  REFR), // water
//  Sphere(800, Cen+Vec(0,-720,-200),Vec(),  Vec(0.106, 0.725, 0.949),  REFR), // water
//  Sphere(800, Cen+Vec(0,-720,-200),Vec(),  Vec(0.110, 0.988, 0.945),  REFR), // waterSphere(800, Cen+Vec(0,-720,-200),Vec(),  Vec(0.110, 0.898, 1.00)*.996,  REFR), // waterSphere(790, Cen+Vec(0,-720,-200),Vec(),  Vec(.4,.3,.04)*.6,    DIFF), // earthSphere(325, Cen+Vec(0,-255,-50), Vec(),  Vec(.4,.3,.04)*.8,       DIFF), // islandSphere(275, Cen+Vec(0,-205,-33), Vec(),  Vec(.02,.3,.02)*.75,      DIFF), // grass
};======================================================================
vista
======================================================================
Vec Cen(50,-20,-860);
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0//     rad       pos                   emis           col     reflSphere(8000, Cen+Vec(0,-8000,-900),Vec(1,.4,.1)*5e-1, Vec(),  DIFF), // sunSphere(1e4,  Cen+Vec(), Vec(0.631, 0.753, 1.00)*3e-1, Vec(1,1,1)*.5,  DIFF), // skySphere(150,  Cen+Vec(-350,0, -100),Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(200,  Cen+Vec(-210,0,-100), Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(145,  Cen+Vec(-210,85,-100),Vec(),  Vec(1,1,1)*.8,  DIFF), // snowSphere(150,  Cen+Vec(-50,0,-100),  Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(150,  Cen+Vec(100,0,-100),  Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(125,  Cen+Vec(250,0,-100),  Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(150,  Cen+Vec(375,0,-100),  Vec(),  Vec(1,1,1)*.3,  DIFF), // mntSphere(2500, Cen+Vec(0,-2400,-500),Vec(),  Vec(1,1,1)*.1,  DIFF), // mnt baseSphere(8000, Cen+Vec(0,-8000,200), Vec(),  Vec(.2,.2,1),    REFR), // waterSphere(8000, Cen+Vec(0,-8000,1100),Vec(),  Vec(0,.3,0),     DIFF), // grassSphere(8   , Cen+Vec(-75, -5, 850),Vec(),  Vec(0,.3,0),     DIFF), // bushSphere(30,   Cen+Vec(0,   23, 825),Vec(),  Vec(1,1,1)*.996, REFR), // ballSphere(30,  Cen+Vec(200,280,-400),  Vec(),  Vec(1,1,1)*.8,  DIFF),   // cloudsSphere(37,  Cen+Vec(237,280,-400),  Vec(),  Vec(1,1,1)*.8,  DIFF),   // cloudsSphere(28,  Cen+Vec(267,280,-400),  Vec(),  Vec(1,1,1)*.8,  DIFF),   // cloudsSphere(40,  Cen+Vec(150,280,-1000),  Vec(),  Vec(1,1,1)*.8,  DIFF),  // cloudsSphere(37,  Cen+Vec(187,280,-1000),  Vec(),  Vec(1,1,1)*.8,  DIFF),  // cloudsSphere(40,  Cen+Vec(600,280,-1100),  Vec(),  Vec(1,1,1)*.8,  DIFF),  // cloudsSphere(37,  Cen+Vec(637,280,-1100),  Vec(),  Vec(1,1,1)*.8,  DIFF),  // cloudsSphere(37,  Cen+Vec(-800,280,-1400),  Vec(),  Vec(1,1,1)*.8,  DIFF), // cloudsSphere(37,  Cen+Vec(0,280,-1600),  Vec(),  Vec(1,1,1)*.8,  DIFF),    // cloudsSphere(37,  Cen+Vec(537,280,-1800),  Vec(),  Vec(1,1,1)*.8,  DIFF),  // clouds};======================================================================
overlap
======================================================================
double D=50;
double R=40;
Sphere spheres[N];// = {//Scene: radius, position, emission, color, materialSphere(150, Vec(50+75,28,62), Vec(1,1,1)*0e-3, Vec(1,.9,.8)*.93, REFR),Sphere(28,  Vec(50+5,-28,62), Vec(1,1,1)*1e1, Vec(1,1,1)*0, DIFF),Sphere(300, Vec(50,28,62), Vec(1,1,1)*0e-3, Vec(1,1,1)*.93, SPEC)
};======================================================================
wada
======================================================================
double R=60;
//double R=120;
double T=30*M_PI/180.;
double D=R/cos(T);
double Z=60;
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0Sphere(1e5, Vec(50, 100, 0),      Vec(1,1,1)*3e0, Vec(), DIFF), // skySphere(1e5, Vec(50, -1e5-D-R, 0), Vec(),     Vec(.1,.1,.1),DIFF),           //grndSphere(R, Vec(50,40.8,62)+Vec( cos(T),sin(T),0)*D, Vec(), Vec(1,.3,.3)*.999, SPEC), //redSphere(R, Vec(50,40.8,62)+Vec(-cos(T),sin(T),0)*D, Vec(), Vec(.3,1,.3)*.999, SPEC), //grnSphere(R, Vec(50,40.8,62)+Vec(0,-1,0)*D,         Vec(), Vec(.3,.3,1)*.999, SPEC), //blueSphere(R, Vec(50,40.8,62)+Vec(0,0,-1)*D,       Vec(), Vec(.53,.53,.53)*.999, SPEC), //backSphere(R, Vec(50,40.8,62)+Vec(0,0,1)*D,      Vec(), Vec(1,1,1)*.999, REFR), //front//   Sphere(R, Vec(50,35,Z)+Vec( cos(T),sin(T),0)*D, Vec(1,1,1)*1e-1, Vec(1,1,1)*.999, SPEC), //red
//   Sphere(R, Vec(50,35,Z)+Vec(-cos(T),sin(T),0)*D, Vec(1,1,1)*1e-1, Vec(1,1,1)*.999, SPEC), //grn
//   Sphere(R, Vec(50,35,Z)+Vec(0,-1,0)*D,           Vec(1,1,1)*1e-1, Vec(1,1,1)*.999, SPEC), //blue
//   Sphere(R, Vec(50,35,Z)+Vec(0,0,-1)*D*1.6,       Vec(1,1,1)*0e-1, Vec(0.275, 0.612, 0.949)*.999, SPEC), //back
//  Sphere(R, Vec(50,40.8,62)+Vec(0,0,1)*D*.2877,          Vec(1,1,1)*0e-1, Vec(1,1,1)*.999, REFR), //front};======================================================================
wada2
======================================================================
//double R=60;
double R=120;     // radius
double T=30*M_PI/180.;
double D=R/cos(T);     //distance
// double D=60;     //distance
// double R=D*sqrt(2);
double Z=62;
Vec C=Vec(0.275, 0.612, 0.949);
Sphere spheres[] = {//Scene: radius, position, emission, color, materialSphere(R, Vec(50,28,Z)+Vec( cos(T),sin(T),0)*D,    C*6e-2,Vec(1,1,1)*.996, SPEC), //redSphere(R, Vec(50,28,Z)+Vec(-cos(T),sin(T),0)*D,    C*6e-2,Vec(1,1,1)*.996, SPEC), //grnSphere(R, Vec(50,28,Z)+Vec(0,-1,0)*D,              C*6e-2,Vec(1,1,1)*.996, SPEC), //blueSphere(R, Vec(50,28,Z)+Vec(0,0,-1)*R*2*sqrt(2./3.),C*0e-2,Vec(1,1,1)*.996, SPEC), //back
//  Sphere(1e5, Vec(50,28,Z)+Vec(0,0,1e5+170),   Vec(1,1,1)*0,Vec(1,1,1)*.996, SPEC), //front
//  Sphere(2*R*2*sqrt(2./3.)-R*2*sqrt(2./3.)/3., Vec(50,28,Z)+Vec(0,0,-R*2*sqrt(2./3.)/3.),   Vec(1,1,1)*0,Vec(1,1,1)*.3333, SPEC), //frontSphere(2*2*R*2*sqrt(2./3.)-R*2*sqrt(2./3.)/3., Vec(50,28,Z)+Vec(0,0,-R*2*sqrt(2./3.)/3.),   Vec(1,1,1)*0,Vec(1,1,1)*.5, SPEC), //front
};======================================================================
forest
======================================================================
Vec tc(0.0588, 0.361, 0.0941);
Vec sc = Vec(1,1,1)*.7;
Sphere spheres[] = {//Scene: radius, position, emission, color, material// center 50 40.8 62// floor 0// back  0
//  Sphere(1e5, Vec(50, 1e5+100, 0),  Vec(1,1,1)*1,Vec(),DIFF), //lite
//  Sphere(1e5, Vec(50, -1e5, 0),  Vec(),Vec(.3,.3,.1),DIFF), //grnd
//  Sphere(1e5, Vec(50, 1e5+100, 0),  Vec(0.761, 0.875, 1.00)*1.3,Vec(),DIFF),
//  //liteSphere(1e5, Vec(50, 1e5+130, 0),  Vec(1,1,1)*1.3,Vec(),DIFF), //liteSphere(1e2, Vec(50, -1e2+2, 47),  Vec(),Vec(1,1,1)*.7,DIFF), //grndSphere(1e4, Vec(50, -30, 300)+Vec(-sin(50*M_PI/180),0,cos(50*M_PI/180))*1e4, Vec(), Vec(1,1,1)*.99,SPEC),// mirr LSphere(1e4, Vec(50, -30, 300)+Vec(sin(50*M_PI/180),0,cos(50*M_PI/180))*1e4,  Vec(), Vec(1,1,1)*.99,SPEC),// mirr RSphere(1e4, Vec(50, -30, -50)+Vec(-sin(30*M_PI/180),0,-cos(30*M_PI/180))*1e4,Vec(), Vec(1,1,1)*.99,SPEC),// mirr FLSphere(1e4, Vec(50, -30, -50)+Vec(sin(30*M_PI/180),0,-cos(30*M_PI/180))*1e4, Vec(), Vec(1,1,1)*.99,SPEC),// mirrSphere(4, Vec(50,6*.6,47),   Vec(),Vec(.13,.066,.033), DIFF),//"tree"Sphere(16,Vec(50,6*2+16*.6,47),   Vec(), tc,  DIFF),//"tree"Sphere(11,Vec(50,6*2+16*.6*2+11*.6,47),   Vec(), tc,  DIFF),//"tree"Sphere(7, Vec(50,6*2+16*.6*2+11*.6*2+7*.6,47),   Vec(), tc,  DIFF),//"tree"Sphere(15.5,Vec(50,1.8+6*2+16*.6,47),   Vec(), sc,  DIFF),//"tree"Sphere(10.5,Vec(50,1.8+6*2+16*.6*2+11*.6,47),   Vec(), sc,  DIFF),//"tree"Sphere(6.5, Vec(50,1.8+6*2+16*.6*2+11*.6*2+7*.6,47),   Vec(), sc,  DIFF),//"tree"
};

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