". return false; These cases can also be easily identified using simple comparisons between the \(t\) values. float tymin = (min.y - r.orig.y) / r.dir.y;
t1z = (B1z - Oz) / Dz It is a kernel operation for BVH-based ray intersection, on which modern high-performance ray-casting algorithms are built. t0z = (B0z - Oz) / Dz
Such a box is called an axis-aligned box and since it will often be a bounding box (or bounding volume) which we also designate by the acronym Computing the intersection of a ray with a AABB is quite simple. t1y = (B1y - Oy) / Dy O’Reilly members experience live online training, plus books, videos, and digital content from tmin = tymin; Show that the local_intersect function for a cube correctly identifies intersections on any face. However intersecting these planes doesn't necessarily mean that these intersecting points lie on the cube (if they don't lie on the cube, obviously the ray doesn't intersect the box). t1x : t1yt0z = (B0z - Oz) / Dz This ray equation is very similar to the line equation. } For instance, if the ray's origin is inside the box (like in the image below), there will be two intersections: one in front of the ray and one behind. Figure 1: equation of a line.
t1x = (B1x - Ox) / Dx if (t1z < tmax) tmax = t1zbool intersect(const Ray &r) { tmin = (max.x - r.orig.x) * invdir.x; if ((tmin > tzmax) || (tzmin > tmax)) Note than when the values for \(t\) are negative, the box is behind the ray.Finally if we apply the same technique to the y and z components we will have at the end of this process a set of six values indicating where the ray intersects the box planes parallel to the x, y and z axis.Note that if the ray is parallel to an axis it won't intersect with the bounding volume plane for this axis (in this case, the line equation for the ray is reduced to the constant \(b\) and there's no solution to equation 1). if (tzmin > tmin) float tmax = (max.x - r.orig.x) / r.dir.x; if ((tmin > tymax) || (tymin > tmax)) if (tymax < tmax) { tmax = (max.x - r.orig.x) * invdir.x; float tzmax = (max.z - r.orig.z) / r.dir.z; Remember that a ray can be expressed using the following parametric form: Where O represents th… float tzmin = (min.z - r.orig.z) / r.dir.z; This can be easily found by operating a simple test on each of the computed \(t\) values. 2004.A Minimal Ray-Tracer: Rendering Simple Shapes (Sphere, Cube, Disk, Plane, etc. The problem at this stage is to find which of these six values correspond to an intersection of the ray with the box (if the ray intersects the box at all. if (tymin > tymax) swap(tymin, tymax); invdir = 1 / dir;
sign[2] = (invdir.z < 0); All we need is to remember that a straight line can be defined using the following analytical equation:In mathematics, the \(m\) term is called the slope (or gradient) and is actually responsible for the orientation of the line and \(b\) corresponds to the point where the line intersects the y-axis. In the eventuality of testing the intersection of the ray against many boxes, we can save some time by pre-computing the inverse direction of the ray in the Ray's constructor and re-using it later in intersect. For Maths Marathon on the Commodore 64, a GameFAQs message board topic titled "Intersecting a(n axis-aligned) cube with a ray? In fact they are the same if you replace \(O\) with \(b\) and \(D\) with \(m\). However intersecting these planes doesn't necessarily mean that these intersecting points lie on the cube (if they don't lie on the cube, obviously the ray doesn't intersect the box). public: float tmin = (min.x - r.orig.x) / r.dir.x; }class Ray bounds[1] = vmax; This though should be handled properly by the compiler which shall return +∞. tmin = (min.x - r.orig.x) / ray.dir.x; } else { When \(t\) is positive, the intersection is in front of the origin of the ray. } When the ray direction is negative, we expect the second block of the if statement to be executed, but in the case where the ray direction is -0, the first block will be executed instead because in IEEE float point, the test -0 = 0 returns true. t0y = (B0y - Oy) / Dy