Binary Space Partitions

The binary space partition (for short, BSP) is a scheme for subdividing the ambient space R into open convex sets (called cells) by hyperplanes in a recursive fashion. Each subdivision step for a cell results in two cells, in which the process may continue, independently of other cells, until a stopping criterion is met. The binary recursion tree, also called BSP-tree, is traditionally used as a data structure in computer graphics for efficient rendering of polyhedral scenes. Each node v of the BSP-tree, except for the leaves, corresponds to a cell Cv R and a partitioning hyperplane Hv. The cell of the root r is Cr D R , and the two children of a node v correspond to Cv \H v and Cv \HC v , where H v and HC v denote the open half-spaces bounded by Hv. Refer to Fig. 1. A binary space partition for a set of n pairwise disjoint (typically polyhedral) objects in R is a BSP where the space is recursively partitioned until each cell intersects at most one object. When the BSP-tree is used as a data structure, every leaf v stores the fragment of at most one object clipped in the cell Cv, and every interior node v stores the fragments of any lower-dimensional objects that lie in Cv \Hv. A BSP for a set of objects has two parameters of interest: the size and the height of the corresponding BSP-tree. Ideally, a BSP partitions space so that each object lies entirely in a single cell or in a cutting hyperplane, yielding a so-called perfect BSP [4]. However, in most cases this is impossible, and the hyperplanes Hv partition some of the input objects into fragments. Assuming that the input objects are k-dimensional, for some k d , the BSP typically stores only k-dimensional fragments, i.e., object parts clipped in leaf cells Cv or in Cv \Hv at interior nodes.