Optimal binary space partitions for orthogonal objects

Abstract

A binary space partition, or BSP, is a scheme for recursively dividing a configuration of objects by hyperplanes until all objects are separated. BSPs are widely used in computer graphics as the underlying data structure for computations such as real-time hidden-surface removal, ray tracing, and solid modeling. In these applications, the computational cost is directly related to the size of the BSP, i.e., the total number of fragments of the objects generated by the partition. Until recently, the question of minimizing the size of BSPs for given inputs had been studied only empirically. We concentrate here on orthogonal objects, a case which arises frequently in practice and deserves special attention. We construct BSPs of linear size for any set of orthogonal line segments in the plane. In three dimensions, BSPs of size O(n 3 2 ) for any set of n mutually orthogonal ine segments or rectangles are constructed. These bounds are optimal and may be contrasted with the Θ( n 2) bound for general polygonal objects in R 3.