问题四十二：怎么用ray tracing画任意圆环片段

42.1 数学推导

引入两个参数theta1、theta2

撞击点P和中心点O连线的向量OP与+X轴的夹角theta在[theta1, theta2]范围内时，才有可能撞击到圆环段。

当然，考虑到theta2可能大于360度，所以，加入另外一种情况：

theta + 2*π <=theta2

theta1的范围[0, 360]

theta2的范围[0,正无穷）

42.2 看C++代码实现

----------------------------------------------tori_part.h ------------------------------------------

tori_part.h

#ifndef TORI_PART_H
#define TORI_PART_H#include "hitable.h"
#include "material.h"
#include "log.h"class tori_part : public hitable
{public:tori_part() {}tori_part(vec3 cen, float ra, float rb, material *m, float t1, float t2) {center = cen;radius_a = ra;radius_b = rb;ma = m;theta1 = t1*M_PI/180;theta2 = t2*M_PI/180;}virtual bool hit(const ray& r, float tmin, float tmax, hit_record& rec) const;vec3 center;float radius_a;float radius_b;material *ma;float theta1, theta2;
};#endif // TORI_PART_H

----------------------------------------------tori_part.cpp ------------------------------------------

tori_part.cpp

#include "tori_part.h"#include <iostream>
using namespace std;bool tori_part::hit(const ray& r, float t_min, float t_max, hit_record& rec) const {
#if TORI_PART_LOG == 1std::cout << "-------------tori_part::hit---1-------------" << endl;
#endif // TORI_PART_LOGvec3 oc = r.origin() - center;float A = dot(oc, oc);float B = 2*dot(oc, r.direction());float C = dot(r.direction(), r.direction());float Rr_square_p = radius_a*radius_a +radius_b*radius_b;float Rr_square_s_square = (radius_a*radius_a - radius_b*radius_b) * (radius_a*radius_a - radius_b*radius_b);float R_square = radius_a*radius_a;float a4 = C*C;float a3 = 2*B*C;float a2 = B*B + 2*A*C - 2*Rr_square_p*C + 4*R_square*r.direction().z()*r.direction().z();float a1 = 2*A*B - 2*Rr_square_p*B + 8*R_square*oc.z()*r.direction().z();float a0 = A*A - 2*Rr_square_p*A + 4*R_square*oc.z()*oc.z() + Rr_square_s_square;float roots[5];roots_quartic_equation2(a4, a3, a2, a1, a0, roots);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;vec3 x = vec3(1.0, 0.0, 0.0);float cos_theta = dot(pc, x) / (pc.length() * x.length());float theta = acos(cos_theta);if (rec.p.y() < center.y()) {theta = 2*M_PI - theta;}if (((theta >= theta1) && (theta <= theta2)) || ((theta + 2*M_PI) <= theta2)) {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[1];

list[0] = new tori_part(vec3(0, 3.2,0), 3.2, 0.2, new lambertian(vec3(0.0, 1.0, 0.0)), 30, 60);

hitable *world = newhitable_list(list,1);

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);

图片输出结果如下：

theta1和theta2分别为30, 60

theta1 和theta2分别为30,150

theta1 和theta2分别为30,210

theta1 和theta2分别为30, 270

theta1 和theta2分别为30, 360

theta1 和theta2分别为30, 380

theta1 和theta2分别为30, 390

搞个组合图：

        hitable *list[7];list[0] = new tori(vec3(0, 3.2, 0), 3.2, 0.2, new lambertian(vec3(0.0, 1.0, 0.0)));list[1] = new tori(vec3(0, 4.8, 0), 1.2, 0.2, new lambertian(vec3(1.0, 0.0, 0.0)));list[2] = new quadratic_cylinder_all(vec3(0, 3.2, 0), 0.2, 0.2, 0.20001, 2.9,new lambertian(vec3(1.0, 0.0, 0.0)), vec3(0, 1, 0), 0);list[3] = new quadratic_cylinder_all(vec3(0, 3.2, 0), 0.2, 0.2, 0.20001, 2.9,new lambertian(vec3(1.0, 0.0, 0.0)), vec3(1, 0, 0), 0);list[4] = new quadratic_cylinder_all(vec3(0, 4.8, 0), 0.2, 0.2, 0.20001, 1.3,new lambertian(vec3(1.0, 0.0, 0.0)), vec3(1, 0, 0), 0);list[5] = new tori_part(vec3(-1.6, 3.2, 0), 1.6, 0.2, new lambertian(vec3(1.0, 0.0, 0.0)), 240, 360);list[6] = new tori_part(vec3(1.6, 3.2, 0), 1.6, 0.2, new lambertian(vec3(1.0, 0.0, 0.0)), 180, 300);hitable *world = new hitable_list(list,7);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);

输出结果如下：（如上位置、长度参数有微调，原参数记不得了）

（by the way, 咱不是开过水果店么，咱店的名字叫做“果园里”，如下图形是本人当时为店名设计的logo）

哦也~看张大图：

等等，图片中心那四段多余的圆弧是什么鬼？

本来高高兴兴，为什么要出现这种事，蓝瘦，香菇

下一章节来解这个问题~~~~~