Space partitioning

In mathematics, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.

Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree.

Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning.

Space partitioning is particularly important in computer graphics, where it is frequently used to organize the objects in a virtual scene. Storing objects in a space-partitioning data structure makes it easy and fast to perform certain kinds of geometry queries — for example, determining whether two objects are close to each other in collision detection, or determining whether a ray intersects an object in ray tracing.

Common space partitioning systems include:

• BSP trees
• Quadtrees
• Octrees
• kd-trees