问题三十六:ray tracing中的Inverse Mapping(1)——球面Inverse Mapping

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:



评论 4
添加红包

请填写红包祝福语或标题

红包个数最小为10个

红包金额最低5元

当前余额3.43前往充值 >
需支付:10.00
成就一亿技术人!
领取后你会自动成为博主和红包主的粉丝 规则
hope_wisdom
发出的红包
实付
使用余额支付
点击重新获取
扫码支付
钱包余额 0

抵扣说明:

1.余额是钱包充值的虚拟货币,按照1:1的比例进行支付金额的抵扣。
2.余额无法直接购买下载,可以购买VIP、付费专栏及课程。

余额充值