Binary space partitioning


In computer science, binary space partitioning is a method for recursively subdividing a 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, where the structure of a BSP tree allows for spatial information about the objects in a scene that is useful in rendering, such as objects being ordered from front-to-back with respect to a viewer at a given location, to be accessed rapidly. Other applications of BSP include: performing geometrical operations with shapes 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 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 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: 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 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. Game engines using BSP trees include the Doom, Quake, 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 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. 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:
  1. Choose a polygon P from the list.
  2. Make a node N in the BSP tree, and add P to the list of polygons at that node.
  3. For each other polygon in the list:
  4. # If that polygon is wholly in front of the plane containing P, move that polygon to the list of nodes in front of P.
  5. # If that polygon is wholly behind the plane containing P, move that polygon to the list of nodes behind P.
  6. # If that polygon is intersected by the plane containing P, split it into two polygons and move them to the respective lists of polygons behind and in front of P.
  7. # If that polygon lies in the plane containing P, add it to the list of polygons at node N.
  8. Apply this algorithm to the list of polygons in front of P.
  9. Apply this algorithm to the list of polygons behind P.
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, 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 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 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,
  1. If the current node is a leaf node, render the polygons at the current node.
  2. Otherwise, if the viewing location V is in front of the current node:
  3. # Render the child BSP tree containing polygons behind the current node
  4. # Render the polygons at the current node
  5. # Render the child BSP tree containing polygons in front of the current node
  6. Otherwise, if the viewing location V is behind the current node:
  7. # Render the child BSP tree containing polygons in front of the current node
  8. # Render the polygons at the current node
  9. # Render the child BSP tree containing polygons behind the current node
  10. Otherwise, the viewing location V must be exactly on the plane associated with the current node. Then:
  11. # Render the child BSP tree containing polygons in front of the current node
  12. # Render the child BSP tree containing polygons behind the current node
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 suitable for the painter's algorithm.

Timeline