Binary space partitioning
In computer science, binary space partitioning (BSP) is a method for space partitioning which recursively subdivides a Euclidean space into two convex sets by using hyperplanes as partitions. This process of subdividing gives rise to a representation of objects within the space in the form of a tree data structure known as a BSP tree. Binary space partitioning was developed in the context of 3D computer graphics in 1969. The structure of a BSP tree is useful in rendering because it can efficiently give spatial information about the objects in a scene, such as objects being ordered  from front-to-back with respect to a viewer at a given location. Other applications of BSP include: performing geometrical operations with shapes (constructive solid geometry) in CAD, collision detection in robotics and 3D video games, ray tracing, and other applications that involve the handling of complex spatial scenes. Overview. Binary space partitioning is a generic process of recursively dividing a scene into two until the partitioning satisfies one or more requirements. It can be seen as a generalization of other spatial tree structures such as "k"-d trees and quadtrees, one where hyperplanes that partition the space may have any  orientation, rather than being aligned with the coordinate axes as they are in "k"-d trees or quadtrees. When used in computer graphics to render scenes composed of planar polygons, the partitioning planes are frequently chosen to coincide with the planes defined by polygons in the scene. The specific choice of partitioning plane and criterion for terminating the partitioning process varies depending on the purpose of the BSP tree. For example, in computer graphics rendering, the scene is divided until each node of the BSP tree contains only polygons that can be rendered in arbitrary order. When back-face culling is  used, each node, therefore, contains a convex set of polygons, whereas when rendering double-sided polygons, each node of the BSP tree contains only polygons in a single plane. In collision detection or ray tracing, a scene may be divided up into primitives on which collision or ray intersection tests are straightforward. Binary space partitioning arose from the computer graphics need to rapidly draw three-dimensional scenes composed of polygons. A simple way to draw such scenes is the painter's algorithm, which produces polygons in order of distance from the viewer, back to front, painting over the background and  previous polygons with each closer object. This approach has two disadvantages: the time required to sort polygons in back-to-front order, and the possibility of errors in overlapping polygons. Fuchs and co-authors showed that constructing a BSP tree solved both of these problems by providing a rapid method of sorting polygons with respect to a given viewpoint (linear in the number of polygons in the scene) and by subdividing overlapping polygons to avoid errors that can occur with the painter's algorithm. A disadvantage of binary space partitioning is that generating a BSP tree can be time-consuming . Typically, it is therefore performed once on static geometry, as a pre-calculation step, prior to rendering or other real-time operations on a scene. The expense of constructing a BSP tree makes it difficult and inefficient to directly implement moving objects into a tree. BSP trees are often used by 3D video games, particularly first-person shooters and those with indoor environments. Game engines using BSP trees include the Doom (id Tech 1), Quake (id Tech 2 variant), GoldSrc and Source engines. In them, BSP trees containing the static geometry of a scene are often used together  with a Z-buffer, to correctly merge movable objects such as doors and characters onto the background scene. While binary space partitioning provides a convenient way to store and retrieve spatial information about polygons in a scene, it does not solve the problem of visible surface determination. Generation. The canonical use of a BSP tree is for rendering polygons (that are double-sided, that is, without back-face culling) with the painter's algorithm. Each polygon is designated with a front side and a backside which could be chosen arbitrarily and only affects the structure of the tree but not  the required result. Such a tree is constructed from an unsorted list of all the polygons in a scene. The recursive algorithm for construction of a BSP tree from that list of polygons is: The following diagram illustrates the use of this algorithm in converting a list of lines or polygons into a BSP tree. At each of the eight steps (i.-viii.), the algorithm above is applied to a list of lines, and one new node is added to the tree. The final number of polygons or lines in a tree is often larger (sometimes much larger) than the original  list, since lines or polygons that cross the partitioning plane must be split into two. It is desirable to minimize this increase, but also to maintain reasonable balance in the final tree. The choice of which polygon or line is used as a partitioning plane (in step 1 of the algorithm) is therefore important in creating an efficient BSP tree. Traversal. A BSP tree is traversed in a linear time, in an order determined by the particular function of the tree. Again using the example of rendering double-sided polygons using the painter's algorithm, to draw a polygon "P  correctly requires that all polygons behind the plane P" lies in must be drawn first, then polygon "P", then finally the polygons in front of "P". If this drawing order is satisfied for all polygons in a scene, then the entire scene renders in the correct order. This procedure can be implemented by recursively traversing a BSP tree using the following algorithm. From a given viewing location "V", to render a BSP tree, Applying this algorithm recursively to the BSP tree generated above results in the following steps: The tree is traversed in linear time and renders the polygons in  a far-to-near ordering ("D1", "B1", "C1", "A", "D2", "B2", "C2", "D3") suitable for the painter's algorithm.