31:T532,md`Bounding volumes play a major role in ray-tracing. If the ray does not intersect the bounding volume of the scene, then it can not intersect any part of the scene. So minimally one must interest the ray with the bounding volume first, and if there is an intersection then only proceed to intersect with the individual objects of the scene.

Bounds trivially define a rectangular box whose two extreme points are (P_min and P_max). The six sides of this box are parallel to three main axis planes. That is why the box is called axis aligned bounding box or AABB. The six sides can be grouped into three sets. The pair

whose x-coordinates of the points are x_min or x_max. They are parallel to YZ-plane (or equivaluently perpendicular to X axis), have (1,0,0) as the normal vector.
whose y-coordinates of the points are y_min or y_max. They are parallel to ZX-plane (or equivaluently perpendicular to Y axis), have (0,1,0) as the normal vector.
whose z-coordinates of the points are z_min and z_max. They are parallel to XY-plane (or equivaluently perpendicular to Z axis) have (0,0,1) as the normal vector.

AABB is the most popular bounding volume for ray tracing.`32:T881,md`The ray intersects a box at upto two points. If we assume the ray parameters of the intersections to be t_near and t_far, then [t_near, t_far] defines the parameter range of all the points of the ray that lie inside the box. It also implies that if the range [t_near, t_far] is null, i.e. t_near< t_far, then there is no ray point of intersection, or in other words, there is no intersection.

The intersection algorithm extends the ray-plane intersection to compute the intersection of the ray with the planes parallel to three pairs of the sides as (t_xmin,t_xmax), (t_ymin,t_ymax), and (t_zmin,t_zmax). As the normals to these planes are vectors parallel to the primary axes, two of the components of the normals are zeros, and thus the ray-plane intersection equation simplifies to a scalar subtraction and a division. So the expression for t's are as follows.
t_xmin = (x_min-x_O)/x_d, t_ymin = (y_min-y_O)/y_d and t_zmin = (z_min-z_O)/z_d,
t_xmax = (x_max-x_O)/x_d, t_ymax = (y_max-y_O)/y_d and t_zmax = (z_max-z_O)/z_d.
Each of these pairs define a scalar range and the range must be non-null. So the algorithm orders the intersection pairs to guaranty validity of the range by swapping them as necessary.
For ex: if (t_xmin > t_xmax) then swap(t_xmin , t_xmax), and so on.
From these ranges the algorithm computes the actual intersection range [t_near,t_far] as
t_near = max(max(t_xmin,t_ymin),t_zmin), and t_far = min(min(t_xmax,t_ymax),t_zmax)
If t_near > t_far then the ray does not intersect the box.`33:[] 2f:{"title":"Bounds and Intersections","cells":[{"id":0,"value":"# Bounds and Intersections","pinned":false,"hidden":false,"mode":"md"},{"id":3,"value":"md`Bounds are important properties of a scene and have many uses in rendering. On this page, we will discuss a method to compute bounds and a few of its applications. \nThe bounds of a 3D object are generally specified as a pair of 3D points: P_min, a point whose x, y, and z components are\nindividually minimum of x, y, and z of every point of the scene along each axis, and P_max, a point whose x, y, and z components are\nindividually maximum. So one way to compute the bounds will be to compute the bounds of each individual object of the scene and then compute the total bound from the individuals.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"md`### Bounds of a Sphere`","pinned":true,"hidden":false,"mode":"ojs"},{"id":11,"value":"md`A sphere is specified by its center coordinate C=(x_c,y_c,z_c) and radius R. So the bounds of a \nsphere is easily computeds as:
\nP_min = (x_c-R,y_c-R,z_c-R), \nand P_max = (x_c+R,y_c+R,z_c+R).\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"md`### Bounds of a Triangle`","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"md`A triangle is specified by its three vertex points P₀=(x₀,y₀,z₀), \nP₁=(x₁,y₁,z₁), P₂=(x₂,y₂,z₂). \nSo the bounds of a triangle can be computed as:
\nP_min = (x_min,y_min,z_min),\nP_max =(x_max,y_max,z_max),
\nwhere x_min = (min(min(x₀,x₁),x₂), x_max = (max(max(x₀,x₁),x₂), and so on.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":37,"value":"md`## Bounding Volumes`","pinned":false,"hidden":false,"mode":"ojs"},{"id":38,"value":"md`### AABB`","pinned":false,"hidden":false,"mode":"ojs"},{"id":35,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":46,"value":"md`#### Ray-AABB Intersection`","pinned":true,"hidden":false,"mode":"ojs"},{"id":45,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":49,"value":"md`### Bounding Sphere`\n\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":51,"value":"md`The next popular bounding volume is sphere. The bounding sphere is easily computed from the bounds of the scene as:
center = 0.5*(P_min + P_max), \nand radius = 0.5*length(P_max - P_min).\n
Intersection of a ray with a bounding sphere is no different from ray-sphere interesection and hence requires no discussion.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":58,"value":"md`### Camera Specification from Bounds`","pinned":false,"hidden":false,"mode":"ojs"},{"id":60,"value":"md`For any arbitrary scene, at a minumum one would like to see what the scene looks like.\nFor a 3D scene a single view is not sufficient. One has to see the scene from different views.\nTo be able to see the whole seen the camera must be placed outside the scene, and the camera must be oriented towards the scene.\nIt is here the bounds of the scene come to rescue. \nA sure way to see the whole scene is to place the camera away from the center of the scene by a diagonal distance, orient it to look at the center of the scene.\nSo here is a sample camera specification derived from the bounds:\n

eye/view point: center + diagonal\nat point: center\nup vector: (0,1,0)\nfov: 54 degrees\nwhere\n  center =  0.5*(P_min + P_max)\n  diagonal = length(P_max - P_min)\n

\nA simple way to derive different views of the whole scene is to generate a new camera \nby rotating the view point around the up vector, while keeping all other camera parameters the same.`","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}