问题四十七：怎么用ray tracing画superellipsoid (2)

网络上有另外一种更为简单的superellipsoid的方程形式：（这种方程，更快些）

数学推导和“问题四十六”类似（应该是更为简单），此处不表。

47.1 看C++代码实现

----------------------------------------------superquadratic.h ------------------------------------------

superquadratic.h

#ifndef SUPERQUADRATIC_H
#define SUPERQUADRATIC_H#include <hitable.h>
#include "material.h"
#include "log.h"class superquadratic : public hitable
{public:superquadratic() {}superquadratic(vec3 cen, float a1, float a2, float a3, float r, float s, float t, material *m) : center(cen), intercept_x(a1), intercept_y(a2), intercept_z(a3), p_r(r), p_s(s), p_t(t), ma(m) {}
/*
f(x,y,z)=(|x/a1|)^r + (|y/a2|)^s + (|z/a3|)^t -1 = 0
*/virtual bool hit(const ray& r, float tmin, float tmax, hit_record& rec) const;vec3 center;float intercept_x, intercept_y, intercept_z;float p_r, p_s, p_t;material *ma;
};#endif // SUPERQUADRATIC_H

----------------------------------------------superquadratic.cpp ------------------------------------------

superquadratic.cpp

#include "superquadratic.h"#include <iostream>
#include <limits>
#include "float.h"
#include "log.h"using namespace std;bool ray_hit_box_q(const ray& r, const vec3& vertex_l, const vec3& vertex_h, float& t_near, float& t_far) {t_near = (numeric_limits<float>::min)();t_far = (numeric_limits<float>::max)();vec3 direction = r.direction();vec3 origin = r.origin();vec3 bl = vertex_l;vec3 bh = vertex_h;float array1[6];if(direction.x() == 0) {if((origin.x() < bl.x()) || (origin.x() > bh.x())) {
#if SUPERQUADRATIC_LOG == 1std::cout << "the ray is parallel to the planes and the origin X0 is not between the slabs. return false" <<endl;
#endif // SUPERQUADRATIC_LOGreturn false;}array1[0] = (numeric_limits<float>::min)();array1[1] = (numeric_limits<float>::max)();}if(direction.y() == 0) {if((origin.y() < bl.y()) || (origin.y() > bh.y())) {
#if SUPERQUADRATIC_LOG == 1std::cout << "the ray is parallel to the planes and the origin Y0 is not between the slabs. return false" <<endl;
#endif // SUPERQUADRATIC_LOGreturn false;}array1[2] = (numeric_limits<float>::min)();array1[3] = (numeric_limits<float>::max)();}if(direction.z() == 0) {if((origin.z() < bl.z()) || (origin.z() > bh.z())) {
#if SUPERQUADRATIC_LOG == 1std::cout << "the ray is parallel to the planes and the origin Z0 is not between the slabs. return false" <<endl;
#endif // SUPERQUADRATIC_LOGreturn false;}array1[4] = (numeric_limits<float>::min)();array1[5] = (numeric_limits<float>::max)();}if((direction.x() != 0) && (direction.y() != 0) && (direction.z() != 0)) {array1[0] = (bl.x()-origin.x())/direction.x();array1[1] = (bh.x()-origin.x())/direction.x();array1[2] = (bl.y()-origin.y())/direction.y();array1[3] = (bh.y()-origin.y())/direction.y();array1[4] = (bl.z()-origin.z())/direction.z();array1[5] = (bh.z()-origin.z())/direction.z();}for (int i=0; i<6; i=i+2){if(array1[i] > array1[i+1]) {float t = array1[i];array1[i] = array1[i+1];array1[i+1] = t;}
#if SUPERQUADRATIC_LOG == 1std::cout << "array1[" << i << "]:" << array1[i] <<endl;std::cout << "array1[" << i+1 << "]:" << array1[i+1] <<endl;
#endif // SUPERQUADRATIC_LOGif(array1[i] >= t_near) {t_near = array1[i];}if(array1[i+1] <= t_far) {t_far = array1[i+1];}if(t_near > t_far) {
#if SUPERQUADRATIC_LOG == 1std::cout << "No.(0=X;2=Y;4=Z):" << i << "  :t_near > t_far. return false" <<endl;
#endif // SUPERQUADRATIC_LOGreturn false;}if(t_far < 0) {
#if SUPERQUADRATIC_LOG == 1std::cout << "No.(0=X;2=Y;4=Z):" << i << "  :t_far < 0. return false" <<endl;
#endif // SUPERQUADRATIC_LOGreturn false;}}return true;
}bool get_superellipsoid_function_and_derivative_q(float a1, float a2, float a3, float er, float es, float et, float xo, float yo, float zo, float xd, float yd, float zd, double t, double& f, double& fd) {double a1_r = double(a1);double a2_r = double(a2);double a3_r = double(a3);double er_r = double(er);double es_r = double(es);double et_r = double(et);double xo_r = double(xo);double yo_r = double(yo);double zo_r = double(zo);double xd_r = double(xd);double yd_r = double(yd);double zd_r = double(zd);double pow_x, pow_y, pow_z, pow_x_d, pow_y_d, pow_z_d;double xd_a1, yd_a2, zd_a3;if ((xo_r+xd_r*t) < 0) { xd_a1 = -xd_r/a1_r; }else { xd_a1 = xd_r/a1_r; }if ((yo_r+yd_r*t) < 0) { yd_a2 = -yd_r/a2_r; }else { yd_a2 = yd_r/a2_r; }if ((zo_r+zd_r*t) < 0) { zd_a3 = -zd_r/a3_r; }else { zd_a3 = zd_r/a3_r; }if ((xo_r+xd_r*t) == 0) {pow_x = 0;pow_x_d = 0;}else {pow_x = pow(fabs(xo_r/a1_r + xd_r*t/a1_r), er_r);pow_x_d = pow(fabs(xo_r/a1_r + xd_r*t/a1_r), er_r-1);}if ((yo_r+yd_r*t) == 0) {pow_y = 0;pow_y_d = 0;}else {pow_y = pow(fabs(yo_r/a2_r + yd_r*t/a2_r), es_r);pow_y_d = pow(fabs(yo_r/a2_r + yd_r*t/a2_r), es_r-1);}if ((zo_r+zd_r*t) == 0) {pow_z = 0;pow_x_d = 0;}else {pow_z = pow(fabs(zo_r/a3_r + zd_r*t/a3_r), et_r);pow_z_d = pow(fabs(zo_r/a3_r + zd_r*t/a3_r), et_r-1);}f = pow_x + pow_y + pow_z - 1;fd = er_r*pow_x_d*xd_a1 + es_r*pow_y_d*yd_a2 + et_r*pow_z_d*zd_a3;if (isnan(f)) {f = f * 1;}if (isnan(fd)) {fd = fd * 1;}return true;
}bool get_roots_by_newton_iteration_q(const ray& r, vec3 c, float a1, float a2, float a3, float er, float es, float et, float x0[8], float (&roots)[2]) {float xo = r.origin().x() - c.x();float yo = r.origin().y() - c.y();float zo = r.origin().z() - c.z();float xd = r.direction().x();float yd = r.direction().y();float zd = r.direction().z();double t_k, t_k1, ft_k, ft_d_k;int j=0;for (int i=0; i<4; i++) {t_k = double(x0[i]);for (int k=0; k<50; k++) {if (!(isnan(t_k))) {get_superellipsoid_function_and_derivative_q(a1, a2, a3, er, es, et, xo, yo, zo, xd, yd, zd, t_k, ft_k, ft_d_k);if ((ft_d_k != 0) && !(isnan(ft_k)) && !(isnan(ft_d_k))) {t_k1 = t_k - ft_k/ft_d_k;if (fabs(t_k1) >= 1) {if (fabs((t_k1 - t_k)/t_k1) < 0.001) {roots[j+1] = float(t_k1);j++;break;}else {t_k = t_k1;}}else {if (fabs(t_k1 - t_k) < 0.001) {roots[j+1] = float(t_k1);j++;break;}else {t_k = t_k1;}}}else {break;}}else {break;}}if (j == 1) {break;}}roots[0] = float(j);if (j == 0) {}return true;
}bool superquadratic::hit(const ray& r, float t_min, float t_max, hit_record& rec) const {
#if SUPERQUADRATIC_LOG == 1std::cout << "-------------superquadratic::hit----------------" << endl;
#endif // SUPERQUADRATIC_LOGvec3 vertex_l[4], vertex_h[4];vertex_l[0] = vec3(center.x()-intercept_x, center.y()-intercept_y, center.z()-intercept_z);vertex_h[0] = vec3(center.x()+intercept_x, center.y()+intercept_y, center.z()+intercept_z);float roots[2], x0[8];float t_near = 0;float t_far = 0;for (int i=0; i<1; i++) {if (ray_hit_box_q(r, vertex_l[i], vertex_h[i], t_near, t_far)) {/*迭代初值，这里提供另一种方式*/x0[0] = t_near;x0[1] = t_far;x0[2] = t_near+1*(t_far - t_near)/4;x0[3] = t_near+3*(t_far - t_near)/4;x0[4] = t_near+1*(t_far - t_near)/8;x0[5] = t_near+7*(t_far - t_near)/8;x0[6] = t_near+3*(t_far - t_near)/8;x0[7] = t_near+5*(t_far - t_near)/8;get_roots_by_newton_iteration_q(r, center, intercept_x, intercept_y, intercept_z, p_r, p_s, p_t, x0, roots);if (roots[0] > 0.0001) {break;}}}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;}}}double nx, ny, nz;float d_a1 = intercept_x;float d_a2 = intercept_y;float d_a3 = intercept_z;vec3 pc;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);pc = rec.p - center;if (pc.x() < 0) {d_a1 = -intercept_x;}if (pc.y() < 0) {d_a2 = -intercept_y;}if (pc.z() < 0) {d_a3 = -intercept_z;}if (pc.x() == 0) {nx = 0;}else {nx = double(p_r)*pow(double(fabs(pc.x()/d_a1)), double(p_r-1))/double(d_a1);}if (pc.y() == 0) {ny = 0;}else {ny = double(p_s)*pow(double(fabs(pc.y()/d_a2)), double(p_s-1))/double(d_a2);}if (pc.z() == 0) {nz = 0;}else {nz = double(p_t)*pow(double(fabs(pc.z()/d_a3)), double(p_t-1))/double(d_a3);}nx = nx/sqrt(nx*nx+ny*ny+nz*nz);ny = ny/sqrt(nx*nx+ny*ny+nz*nz);nz = nz/sqrt(nx*nx+ny*ny+nz*nz);rec.normal = unit_vector(vec3(float(nx), float(ny), float(nz)));if(dot(r.direction(), rec.normal) > 0) {rec.normal = - rec.normal;}rec.mat_ptr = ma;rec.u = -1.0;rec.v = -1.0;return true;}}return false;}else {return false;}return false;
}

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

main.cpp

        hitable *list[1];list[0] = newsuperquadratic(vec3(0, 3, 0), 3, 4.5, 3, 0.5, 0.5, 0.5, new lambertian(vec3(0.0,1.0, 0.0)));hitable *world = newhitable_list(list,1);vec3 lookfrom(20, 20, 20);vec3 lookat(0, 3, 0);float dist_to_focus =(lookfrom - lookat).length();float aperture = 0.0;camera cam(lookfrom,lookat, vec3(0,1,0), 30, float(nx)/float(ny), aperture, 0.7*dist_to_focus);

输出图片如下：

三个参数一起变

r=0.5, s=0.5, t=0.5

r=1, s=1, t=1

r=1.5, s=1.5, t=1.5

r=2, s=2, t=2

r=4, s=4, t=4

r=6, s=6, t=6

r=10, s=10, t=10

r=20, s=20, t=20

r=500, s=500, t=500

r=1000, s=1000, t=1000

r=2, t=2,只改变s

r=2, s=0.01, t=2

r=2, s=0.05, t=2

r=2, s=0.1, t=2

r=2, s=0.25, t=2

r=2, s=0.5, t=2

r=2, s=1, t=2

r=2, s=4, t=2

r=2, s=10, t=2

r=2, s=20, t=2

r=2, s=40, t=2

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

main.cpp

        hitable *list[1];list[0] = new superquadratic(vec3(0, 3, 0), 3, 4.5, 3, 0.5, 5, 0.5, new lambertian(vec3(0.0, 1.0, 0.0)));hitable *world = new hitable_list(list,1);vec3 lookfrom(10, 20, 20);vec3 lookat(0, 3, 0);float dist_to_focus = (lookfrom - lookat).length();float aperture = 0.0;camera cam(lookfrom, lookat, vec3(0,1,0), 30, float(nx)/float(ny), aperture, 0.7*dist_to_focus);

输出图片如下：

r=0.5, s=5, t=0.5

r=0.5, s=10, t=0.5

r=0.5, s=20, t=0.5

r=0.5, s=30, t=0.5

47.2 问题分析

前面之所以要选四个box，而不是一个或者2两个，原因就是对于某些像素点，迭代初始值离真实值太远，导致迭代无法收敛，导致无法求得方程的解，导致解少了。

如下这组图片，可以看出初始值对图片效果的影响（这组图片是“问题四十七”的）：

只有最大的box，且只用t_near时的图片：

4个box，且只用t_near时的图片：

4个box，且用t_near和t_far时的图片：

总结可能出现的问题：

1，像素点值溢出：除数为0；pow()函数的三种特殊情况：基数为负，指数为非整数；基数为0；计算值超出数据类型范围。

2，少根：迭代初值离真实值距离太远，导致迭代不收敛；误差太小，无法迭代到该精确度。

3，多根：误差太大，不是根的点也成了根。