本文介绍了一种使用迭代法求解超级双曲面方程的方法，并通过C++代码实现了单页和双页超级双曲面的渲染。文章详细讨论了求解过程中遇到的问题及解决方法。
48.1 数学推导
Superhyperboloid的方程和“问题四十六”中描述的superellipsoid的如下方程非常接近。
到获得迭代格式，超级双曲面和超级椭圆面的推导是相似的，此处不表。
超级双曲面和超级椭圆面的差异之处在于：
超级双曲面是不封闭的面，所以，我们在y轴方向对曲面的范围进行限制。
其一：由于这个“限制”，导致迭代得到的解也必须在这个“限制”内。
其二：由于方程的改变，求迭代初值的几个box也有所变化。
其三：对迭代初值的选取要求更为苛刻（一不小心就少根了）。
48.2 看C++代码实现
----------------------------------------------superhyperboloid.h ------------------------------------------
superhyperboloid.h
#ifndef SUPERHYPERBOLOID_H
#define SUPERHYPERBOLOID_H
#include <hitable.h>
#include "material.h"
#include "log.h"
class superhyperboloid : public hitable
{
public:
superhyperboloid() {}
superhyperboloid(vec3 cen, float a1, float a2, float a3, float e1, float e2, float s1, float s2, float hy, int in, float tol, material *m) :
center(cen), intercept_x(a1), intercept_y(a2), intercept_z(a3), p_e1(e1), p_e2(e2),
sign1(s1), sign2(s2), half_y(hy), initial_number(in), tolerance(tol), ma(m) {}
/*
f(x,y,z)=( (x/a1)^(2/e2) + s2*(z/a3)^(2/e2) )^(e2/e1) + s1*(y/a2)^(2/e1) -1 = 0
in: initial number//迭代初值个数
tol: tolerance
s1,s2: 1, 1: superellipsoid
s1,s2:-1, 1: superhyperboloids of one sheet
s1,s2:-1,-1: superhyperboloids of two sheets
hy: half height of the surface in y-direction
*/
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_e1, p_e2;
float sign1, sign2;
float half_y;
int initial_number;
float tolerance;
material *ma;
};
#endif // SUPERHYPERBOLOID_H
----------------------------------------------superhyperboloid.cpp ------------------------------------------
superhyperboloid.cpp
#include "superhyperboloid.h"
#include <iostream>
#include <limits>
#include "float.h"
#include "log.h"
using namespace std;
bool ray_hit_box_h(const ray& r, const vec3& vertex_l, const vec3& vertex_h, float& t_near, float& t_far) {//这个函数和之前的一样，不解释
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 SUPERHYPERBOLOID_LOG == 1
std::cout << "the ray is parallel to the planes and the origin X0 is not between the slabs. return false" <<endl;
#endif // SUPERHYPERBOLOID_LOG
return 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 SUPERHYPERBOLOID_LOG == 1
std::cout << "the ray is parallel to the planes and the origin Y0 is not between the slabs. return false" <<endl;
#endif // SUPERHYPERBOLOID_LOG
return 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 SUPERHYPERBOLOID_LOG == 1
std::cout << "the ray is parallel to the planes and the origin Z0 is not between the slabs. return false" <<endl;
#endif // SUPERHYPERBOLOID_LOG
return 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 SUPERHYPERBOLOID_LOG == 1
std::cout << "array1[" << i << "]:" << array1[i] <<endl;
std::cout << "array1[" << i+1 << "]:" << array1[i+1] <<endl;
#endif // SUPERHYPERBOLOID_LOG
if(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 SUPERHYPERBOLOID_LOG == 1
std::cout << "No.(0=X;2=Y;4=Z):" << i << " :t_near > t_far. return false" <<endl;
#endif // SUPERHYPERBOLOID_LOG
return false;
}
if(t_far < 0) {
#if SUPERHYPERBOLOID_LOG == 1
std::cout << "No.(0=X;2=Y;4=Z):" << i << " :t_far < 0. return false" <<endl;
#endif // SUPERHYPERBOLOID_LOG
return false;
}
}
if (t_near != t_near) {
t_near = t_near * 1;
}
return true;
}
bool get_superellipsoid_function_and_derivative_h(float a1, float a2, float a3, float e1, float e2, float s1, float s2, 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 e1_r = double(e1);
double e2_r = double(e2);
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), (2/e2_r));
pow_x_d = pow(fabs(xo_r/a1_r + xd_r*t/a1_r), ((2/e2_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), (2/e1_r));
pow_y_d = pow(fabs(yo_r/a2_r + yd_r*t/a2_r), ((2/e1_r)-1));
}
if ((zo_r+zd_r*t) == 0) {
pow_z = 0;
pow_z_d = 0;
}
else {
pow_z = pow(fabs(zo_r/a3_r + zd_r*t/a3_r), (2/e2_r));
pow_z_d = pow(fabs(zo_r/a3_r + zd_r*t/a3_r), ((2/e2_r)-1));
}
if((pow_x+s2*pow_z) == 0) {
f = pow_y - 1;
fd = (2/e1_r)*pow_y_d*yd_a2;
}
elseif ((pow_x+s2*pow_z) < 0) {
//对于双页双曲面，s2为负，所以这里需要判断一下
f = -pow(-(pow_x+s2*pow_z), (e2_r/e1_r)) + s1*pow_y - 1;
fd = (2/e1_r)*(-pow(-(pow_x+s2*pow_z), ((e2_r/e1_r)-1))
*(pow_x_d*xd_a1+s2*pow_z_d*zd_a3)+ s1*pow_y_d*yd_a2);
}
else {
f = pow(pow_x+s2*pow_z, (e2_r/e1_r)) + s1*pow_y - 1;
fd = (2/e1_r)*(pow(pow_x+s2*pow_z, ((e2_r/e1_r)-1))
*(pow_x_d*xd_a1+s2*pow_z_d*zd_a3) + s1*pow_y_d*yd_a2);
}
return true;
}
bool get_roots_by_newton_iteration_h(const ray& r, vec3 c, float a1, float a2, float a3, float e1, float e2, float s1, float s2, int in, float tol, float *x0, float (&roots)[3]) {
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, in_r;
if (in > int(x0[0])) {
in_r = int(x0[0]);
}
else {
in_r = in;
}
for (int i=1; i<=in_r; i++) {
t_k = double(x0[i]);
for (int k=0; k<50; k++) {
if (!(isnan(t_k))) {
get_superellipsoid_function_and_derivative_h(a1, a2, a3, e1, e2, s1, s2, 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) < tol) {
if ((t_k1 >= x0[1]) && (t_k1 <= x0[in_r])) {
/*传进该函数的x0序列是从t_near到t_far排序好的，由于y轴方向的高度限制，所求的解必须在（t_near, t_far）范围内。By the way, 我们在这里值求一个解，这里求得的解不确定是大根还是小根。*/
roots[j+1] = float(t_k1);
j++;
break;
}
else {
break;
}
}
else {
t_k = t_k1;
}
}
else {
break;
}
}
else {
break;
}
}
if (j == 1) {
break;
}
}
roots[0] = float(j);
return true;
}
bool superhyperboloid::hit(const ray& r, float t_min, float t_max, hit_record& rec) const {
#if SUPERHYPERBOLOID_LOG == 1
std::cout << "-------------superhyperboloid::hit----------------" << endl;
#endif // SUPERHYPERBOLOID_LOG
vec3 vertex_l[1], vertex_h[1];
float intercept_hyper_x, intercept_hyper_z;
float pow_y = pow((half_y/intercept_y), (2/p_e1));
intercept_hyper_x = intercept_x*pow((pow_y+1), (p_e1/2));
intercept_hyper_z = intercept_z*pow((pow_y+1), (p_e1/2));
/*y轴方向的半长度为half_y。在方程中，设y=half_y, x=0，可以求得intercept_hyper_z；在方程中，设y=half_y, z=0，可以求得intercept_hyper_x 。这里的intercept_hyper_x, half_y, intercept_hyper_z就相当于“问题四十六”中的a1, a2, a3*/
vertex_l[0] = vec3(center.x()-intercept_hyper_x, center.y()-half_y, center.z()-intercept_hyper_z);
vertex_h[0] = vec3(center.x()+intercept_hyper_x, center.y()+half_y, center.z()+intercept_hyper_z);
float roots[3] = {0.0, -1.0, -1.0};
float x0[initial_number+1];
float t_near = 0;
float t_far = 0;
if (ray_hit_box_h(r, vertex_l[0], vertex_h[0], t_near, t_far)) {
for (int i=0; i<initial_number; i++) {
x0[i+1] = t_near + i*(t_far - t_near)/(initial_number-1);
/*这里我们将t_near, t_far进行若干等分得到若干个x0的值作为迭代初值（序列）*/
}
x0[0] = float(initial_number);
get_roots_by_newton_iteration_h(r, center, intercept_x, intercept_y, intercept_z, p_e1, p_e2, sign1, sign2, initial_number, tolerance, x0, roots);
}
else {
return false;
}
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, pow_x, pow_z, pow_x_d, pow_z_d, pow_y_d, pow_x_z_d;
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);
if (((sign1 == -1)&&(sign2 == 1)) || ((sign1 == -1)&&(sign2 == -1))) {
if (!(((rec.p.y()-center.y()) >= -half_y)
&& ((rec.p.y()-center.y()) <= half_y))) {
/*其实这句是多余的，因为在迭代函数中已经用t_near, t_far对解进行了限制*/
continue;
}
}
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) {
pow_x = 0;
pow_x_d = 0;
}
else {
pow_x = pow(double(fabs(pc.x()/d_a1)), double(2/p_e2));
pow_x_d = pow(double(fabs(pc.x()/d_a1)), double(2/p_e2-1));
}
if (pc.y() == 0) {
pow_y_d = 0;
}
else {
pow_y_d = pow(double(fabs(pc.y()/d_a2)), double(2/p_e1-1));
}
if (pc.z() == 0) {
pow_z = 0;
pow_z_d = 0;
}
else {
pow_z = pow(double(fabs(pc.z()/d_a3)), double(2/p_e2));
pow_z_d = pow(double(fabs(pc.z()/d_a3)), double(2/p_e2-1));
}
if ((pow_x+sign2*pow_z) == 0) {
pow_x_z_d = 0;
}
if ((pow_x+sign2*pow_z) < 0) {//当sign2为负时，需要判断一下
pow_x_z_d = -pow(-(pow_x+sign2*pow_z), double(p_e2/p_e1-1));
}
else {
pow_x_z_d = pow(pow_x+sign2*pow_z, double(p_e2/p_e1-1));
}
nx = double(2/p_e1) * pow_x_z_d * pow_x_d / d_a1;
ny = double(2/p_e1) *sign1* pow_y_d / d_a2;
nz = double(2/p_e1) * pow_x_z_d * pow_z_d / d_a3;
if (isnan(nx)) {
nx = nx * 1;
}
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;
}
48.2.1 Superhyperboloid one sheet
----------------------------------------------main.cpp ------------------------------------------
main.cpp
hitable *list[1];
list[0] = new superhyperboloid(vec3(0, 3, 0), 3, 4.5, 3, 3, 0.1, -1, 1, 4.5, 16, 0.001, new lambertian(vec3(0.0, 1.0, 0.0)));
/*half_y为4.5；迭代初值个数为16，只有将t_near, t_far等分为这么多份，才能保证图片基本不缺块，这就是之前说的“苛刻”的地方；误差为0.001*/
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：
e1=0.1, e2=0.1
e1=1, e2=0.1
e1=2, e2=0.1
e1=3, e2=0.1
e1=0.1, e2=1
e1=1, e2=1
e1=2, e2=1
e1=3, e2=1
e1=0.1, e2=2
e1=1, e2=2
e1=2, e2=2
e1=3, e2=2
e1=0.1, e2=3
e1=1, e2=3
e1=2, e2=3
e1=3, e2=3
48.2.2 Superhyperboloid two sheet
----------------------------------------------main.cpp ------------------------------------------
main.cpp
hitable *list[1];
list[0] = newsuperhyperboloid(vec3(0, 3, 0), 3, 4.5, 3, 0.1, 0.1, -1, -1,4.5, 16, 0.001, new lambertian(vec3(0.0, 1.0, 0.0)));
/*half_y为4.5；迭代初值个数为16，只有将t_near, t_far等分为这么多份，才能保证图片基本不缺块，这就是之前说的“苛刻”的地方；误差为0.001*/
hitable *world = newhitable_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：
双页双曲面的实现还有问题，估计迭代初值和误差等等还需要调。对于学习进度的考虑，在此不调了。
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