问题三十九：怎么用ray tracing画圆环

39.1 数学推导

书上这么说：

39.2 看C++代码实现

----------------------------------------------tori.h------------------------------------------

tori.h

#ifndef TORI_H
#define TORI_H#include <hitable.h>class tori : public hitable
{public:tori() {}tori(vec3 cen, float ra, float rb, material *m) : center(cen), radius_a(ra), radius_b(rb), ma(m) {}virtual bool hit(const ray& r, float tmin, float tmax, hit_record& rec) const;vec3 center;float radius_a;float radius_b;material *ma;
};#endif // TORI_H

----------------------------------------------tori.cpp------------------------------------------

tori.cpp

#include "tori.h"
#include <iostream>
using namespace std;bool tori::hit(const ray& r, float t_min, float t_max, hit_record& rec) const {vec3 oc = r.origin() - center;float a4 = dot(r.direction(), r.direction()) * dot(r.direction(), r.direction());float a3 = 4*(dot(oc, r.direction())) * dot(r.direction(), r.direction());float a2 = 2*dot(r.direction(), r.direction()) * (dot(oc, oc) - (radius_a*radius_a+radius_b*radius_b))+ 2*(dot(oc, r.direction()))- 4*radius_a*radius_a*r.direction().z()*r.direction().z();float a1 = 4*dot(oc, r.direction())*(dot(oc, oc) - (radius_a*radius_a+radius_b*radius_b))- 8*radius_a*radius_a*oc.z()*r.direction().z();float a0 = (dot(oc, oc) - (radius_a*radius_a+radius_b*radius_b))                    - 4*radius_a*radius_a*(radius_b*radius_b-oc.z()*oc.z());float *roots = roots_quartic_equation(a4, a3, a2, a1, a0);float temp;if (roots[0] > 0.0001) {for (int i=1; i<int(roots[0]); i++) {for (int j=i+1; j<int(roots[0])+1; j++) {if (roots[i] > roots[j]) {temp = roots[i];roots[i] = roots[j];roots[j] = temp;}}}for (int k=1; k<int(roots[0])+1; k++) {if (roots[k] < t_max && roots[k] > t_min) {rec.t = roots[k];rec.p = r.point_at_parameter(rec.t);vec3 pc = rec.p - center;float nx = 4*pc.x()*pc.x()*pc.x() + 4*pc.x()*(pc.y()*pc.y()+pc.z()*pc.z()-(radius_a*radius_a+radius_b*radius_b));float ny = 4*pc.y()*pc.y()*pc.y() + 4*pc.y()*(pc.x()*pc.x()+pc.z()*pc.z()-(radius_a*radius_a+radius_b*radius_b));float nz = 4*pc.z()*pc.z()*pc.z() + 4*pc.z()*(pc.x()*pc.x()+pc.y()*pc.y()+radius_a*radius_a-radius_b*radius_b);rec.normal = unit_vector(vec3(nx, ny, nz));if(dot(r.direction(), rec.normal) > 0) {rec.normal = - rec.normal;}rec.mat_ptr = ma;rec.u = -1.0;rec.v = -1.0;delete [] roots;return true;}}}delete [] roots;return false;
}

----------------------------------------------main.cpp------------------------------------------

main.cpp

        hitable *list[2];list[0] = new sphere(vec3(0.0,-100,0), 100, new lambertian(vec3(0.8, 0.8, 0.0)));list[1] = new tori(vec3(0, 3.0, 0), 2.5, 0.5, new lambertian(vec3(1.0, 0.5, 0.5)));hitable *world = new hitable_list(list,2);vec3 lookfrom(0, 3, 5);vec3 lookat(0, 3, 0);float dist_to_focus = (lookfrom - lookat).length();float aperture = 0.0;camera cam(lookfrom, lookat, vec3(0,1,0), 80, float(nx)/float(ny), aperture, 0.7*dist_to_focus);

输出图片如下：

我只是想问问：这是什么鬼？？？

39.3 修改书上错误

查来查去，算来算去。发现是书上对一元四次方程的系数算错了。坑爹啊！！！

书上给的是这个：

实际应该是这个：

对应代码修改如下：

float a4 = dot(r.direction(), r.direction()) * dot(r.direction(),r.direction());

float a3 = 4*(dot(oc, r.direction())) *dot(r.direction(), r.direction());

float a2 = 2*dot(r.direction(), r.direction()) * (dot(oc, oc) -(radius_a*radius_a+radius_b*radius_b))

+ 4*(dot(oc,r.direction()))*(dot(oc, r.direction()))

+4*radius_a*radius_a*r.direction().z()*r.direction().z();

float a1 = 4*dot(oc, r.direction())*(dot(oc,oc) - (radius_a*radius_a+radius_b*radius_b))

+8*radius_a*radius_a*oc.z()*r.direction().z();

float a0 = (dot(oc, oc) - (radius_a*radius_a+radius_b*radius_b))*(dot(oc, oc) - (radius_a*radius_a+radius_b*radius_b))

-4*radius_a*radius_a*(radius_b*radius_b-oc.z()*oc.z());

修改后输出图片如下：

哦啦~第一张圆环图片应该正常了……

再来一张：

        hitable *list[2];list[0] = new tori(vec3(0, 3.4, 0), 3.2, 0.2, new lambertian(vec3(0.0, 1.0, 0.0)));list[1] = new tori(vec3(0, 4.9, 0), 1.3, 0.2, new lambertian(vec3(1.0, 0.0, 0.0)));hitable *world = new hitable_list(list,2);vec3 lookfrom(0, 3, 5);vec3 lookat(0, 3, 0);float dist_to_focus = (lookfrom - lookat).length();float aperture = 0.0;camera cam(lookfrom, lookat, vec3(0,1,0), 80, float(nx)/float(ny), aperture, 0.7*dist_to_focus);

输出图片如下：

等等……等等……中间那个白色夹杂着红色的圈是什么鬼？？？