36.1 球面Inverse Mapping
36.1.1 数学推导
球面(球心在(X0, Y0, Z0),半径为R)上任意一点P的坐标(Xi, Yi, Zi)的参数表示为:
Xi = X0 + R*sinα*sinβ
Yi = Y0 + R*cosα
Zi = Z0 + R*sinα*cosβ
所以,P点的法向量Sn的坐标(Xn, Yn, Zn)可以表示为:
Xn = Xi – X0 = R*sinα*sinβ
Yn = Yi – Y0 = R*cosα
Zn = Zi – Z0 = R*sinα*cosβ
标准化之后,Sn = (sinα*sinβ,cosα,sinα*cosβ)
另外,Y、Z轴方向的单位向量Sp、Se分别为:
Sp = (0,1,0)
Se = (0,0,1)
如上图,dot(Sn,Sp)=|Sn||Sp|*cosα
所以,cos α=dot(Sn, Sp)/(|Sn||Sp|)
有因为,Sn、Sp为单位向量,
所以,cos α=dot(Sn, Sp)
有因为,α= π-φ ,cos(π-φ)= (- cosφ)= cos α= dot(Sn, Sp)
所以,cosφ = -dot(Sn, Sp)
所以,φ = arccos(- dot(Sn, Sp))
由于φ的范围为[0, π]
所以,φ/π的范围为[0, 1]
36.1.2 看C++代码实现
----------------------------------------------sphere.cpp ------------------------------------------
sphere.cpp
#include "sphere.h"
#include <iostream>
using namespace std;
bool sphere::hit(const ray& r, float t_min, float t_max, hit_record& rec) const {
#if SPHERE_LOG == 1
std::cout << "-------------sphere::hit----------------" << endl;
#endif // SPHERE_LOG
vec3 oc = r.origin() - center;
float a = dot(r.direction(), r.direction());
float b = 2.0 * dot(oc, r.direction());
float c = dot(oc, oc) - radius*radius;
float discriminant = b*b - 4*a*c;
if (discriminant > 0) {
float temp = (-b - sqrt(discriminant)) / (2.0*a);
if (temp < t_max && temp > t_min) {
rec.t = temp;
rec.p = r.point_at_parameter(rec.t);
rec.normal = unit_vector((rec.p - center) / radius);
rec.mat_ptr = ma;
vec3 pole = vec3(0, 1, 0);
vec3 equator = vec3(0, 0, 1);
float u, v;
float phi = acos(-dot(rec.normal, pole));
v = phi / M_PI;
float theta = acos((dot(equator, rec.normal)) / sin(phi)) / (2*M_PI);
if (dot(cross(pole, equator), rec.normal) > 0) {
u = theta;
}
else {
u = 1 - theta;
}
rec.u = u;
rec.v = v;
return true;
}
temp = (-b + sqrt(discriminant)) / (2.0*a);
if (temp < t_max && temp > t_min) {
rec.t = temp;
rec.p = r.point_at_parameter(rec.t);
rec.normal = unit_vector((rec.p - center) / radius);
rec.mat_ptr = ma;
vec3 pole = vec3(0, 1, 0);
vec3 equator = vec3(0, 0, 1);
float u, v;
float phi = acos(-dot(rec.normal, pole));
v = phi / M_PI;
float theta = acos((dot(equator, rec.normal)) / sin(phi)) / (2*M_PI);
if (dot(cross(pole, equator), rec.normal) > 0) {
u = theta;
}
else {
u = 1 - theta;
}
rec.u = u;
rec.v = v;
return true;
}
}
// std::cout << "-------------sphere::hit---3-------------" << endl;
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 sphere(vec3(0, 2.5, 0), 2.5, new lambertian(vec3(0.8, 0.8, 0.0)));
hitable *world = new hitable_list(list,2);
vec3 lookfrom(0, 2.5, 20);
vec3 lookat(0, 2.5, 0);
float dist_to_focus = (lookfrom - lookat).length();
float aperture = 0.0;
camera cam(lookfrom, lookat, vec3(0,1,0), 20, float(nx)/float(ny), aperture, 0.7*dist_to_focus);
输出图片如下:
vec3 lookfrom(0, 2.5, 20);
vec3 lookfrom(0, 2.5, -20);
uv原图: