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In computer science , binary space partitioning BSP is a method for recursively subdividing a space into convex sets by hyperplanes. This subdivision gives rise to a representation of objects within the space by means of a tree data structure known as a BSP tree. Binary space partitioning was developed in the context of 3D computer graphics ,   where the structure of a BSP tree allows spatial information about the objects in a scene that is useful in rendering , such as their ordering from front-to-back with respect to a viewer at a given location, to be accessed rapidly.
Other applications include performing geometrical operations with shapes constructive solid geometry in CAD ,  collision detection in robotics and 3D video games , ray tracing and other computer applications that involve handling of complex spatial scenes. 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 generalisation 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 render 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: 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 realtime 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. 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. 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 back side which could be chosen arbitrarily and only affects the structure of the tree but not the required result.
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. 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. 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.
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. From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.
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Following the steps of the algorithm above, We choose a line, A, from the list and, We split the remaining lines in the list into those in front of A i. We first process the lines in front of A in steps ii—v , We now apply the algorithm to the list of lines in front of A containing B2, C2, D2. We choose a line, B2, add it to a node and split the rest of the list into those lines that are in front of B2 D2 , and those that are behind it C2, D3.
Choose a line, D2, from the list of lines in front of B2 and A. It is the only line in the list, so after adding it to a node, nothing further needs to be done.
We are done with the lines in front of B2, so consider the lines behind B2 C2 and D3. Choose one of these C2 , add it to a node, and put the other line in the list D3 into the list of lines in front of C2.
Now look at the list of lines in front of C2. There is only one line D3 , so add this to a node and continue. We have now added all of the lines in front of A to the BSP tree, so we now start on the list of lines behind A. Choosing a line B1 from this list, we add B1 to a node and split the remainder of the list into lines in front of B1 i. D1 , and lines behind B1 i. Processing first the list of lines in front of B1, D1 is the only line in this list, so add this to a node and continue.
Looking next at the list of lines behind B1, the only line in this list is C1, so add this to a node, and the BSP tree is complete.