【Ray Tracing from Ground Up】Ray-Object Interserctions

1.Bounding Boxes

    如下，没有和box 相交的线在 的 interval 和的interval 没有交集。相反，有相交的线的 两个interval 之间是交集的

bool  Box::hit(const Ray& ray, double& tmin, ShadeRec& sr) const {double ox = ray.o.x; double oy = ray.o.y; double oz = ray.o.z;double dx = ray.d.x; double dy = ray.d.y; double dz = ray.d.z;double tx_min, ty_min, tz_min;double tx_max, ty_max, tz_max;double a = 1.0 / dx;if (a >= 0) {tx_min = (x0 - ox) * a;tx_max = (x1 - ox) * a;}else {tx_min = (x1 - ox) * a;tx_max = (x0 - ox) * a;}double b = 1.0 / dy;if (b >= 0) {ty_min = (y0 - oy) * b;ty_max = (y1 - oy) * b;}else {ty_min = (y1 - oy) * b;ty_max = (y0 - oy) * b;}double c = 1.0 / dz;if (c >= 0) {tz_min = (z0 - oz) * c;tz_max = (z1 - oz) * c;}else {tz_min = (z1 - oz) * c;tz_max = (z0 - oz) * c;}double t0, t1;int face_in, face_out;// find largest entering t valueif (tx_min > ty_min) {t0 = tx_min;face_in = (a >= 0.0) ? 0 : 3;}else {t0 = ty_min;face_in = (b >= 0.0) ? 1 : 4;}if (tz_min > t0) {t0 = tz_min;face_in = (c >= 0.0) ? 2 : 5;}// find smallest exiting t valueif (tx_max < ty_max) {t1 = tx_max;face_out = (a >= 0.0) ? 3 : 0;}else {t1 = ty_max;face_out = (b >= 0.0) ? 4 : 1;}if (tz_max < t1) {t1 = tz_max;face_out = (c >= 0.0) ? 5 : 2;}if (t0 < t1 && t1 > kEpsilon) {  // condition for a hitif (t0 > kEpsilon) {tmin = t0;  // ray hits outside surfacesr.normal = get_normal(face_in);}else {tmin = t1;// ray hits inside surfacesr.normal = get_normal(face_out);}sr.local_hit_point = ray.o + tmin * ray.d;return (true);}elsereturn (false);}

2. Triangles

三角形的定义

三角形的光线碰撞检测，分为两步：①光线是否hit 三角形所在的平面 ② hit point 是否在三角形以内。三角形所在平面任意一点p 可以表示为，带入，，最后整理得到。 由此，

   ==> ,  根据 克莱姆 法则，可解得

bool Triangle::hit(const Ray& ray, double& tmin, ShadeRec& sr) const {double a = v0.x - v1.x, b = v0.x - v2.x, c = ray.d.x, d = v0.x - ray.o.x; double e = v0.y - v1.y, f = v0.y - v2.y, g = ray.d.y, h = v0.y - ray.o.y;double i = v0.z - v1.z, j = v0.z - v2.z, k = ray.d.z, l = v0.z - ray.o.z;double m = f * k - g * j, n = h * k - g * l, p = f * l - h * j;double q = g * i - e * k, s = e * j - f * i;double inv_denom  = 1.0 / (a * m + b * q + c * s);double e1 = d * m - b * n - c * p;double beta = e1 * inv_denom;if (beta < 0.0) return (false);double r = r = e * l - h * i;double e2 = a * n + d * q + c * r;double gamma = e2 * inv_denom;if (gamma < 0.0 ) return (false);if (beta + gamma > 1.0)return (false);double e3 = a * p - b * r + d * s;double t = e3 * inv_denom;if (t < kEpsilon) return (false);tmin = t;sr.normal = normal;  sr.local_hit_point = ray.o + t * ray.d;return (true);}  

3.Other Objects

  圆形由中心c，半径r，法线n确定 ,同理首先是判断是不是在圆形所在的平面，然后判断和圆心的距离，既可以看出是否和圆形有相交。 因为开方运算耗时，所以直接判断平方的大小