Stabbing Orthogonal Objects in 3-Space

David M. Mount Fan-Tao Pu Department of Computer Science University of Maryland College Park, MD 20742 fmount,[email protected] 1 Motivation Computer graphics is the source of many interesting and challenging applications for the design of geometric algorithms and data structures. Applications in global illumination simulation and radiosity [7] have motivated our study of problems involving lines in 3-space. In particular we are interested in geometric data structures for answering visibility range queries. Given a 3-dimensional scene de ned by a set of polygonal patches, we wish to preprocess the scene to answer queries involving the set of patches of the scene that are visible from a given range of points over a given range of viewing directions. Queries of this form are central to global illumination simulation. This has led us to the study of data structures for storing and accessing subdivisions over directed-line space, where each point in this space is associated with a directed line in 3-space. In the search for these data structures, we have come across an important problem. Our solution to this problem shows that choices for the data structures used to store the 3-dimensional scene may have a considerable impact on the design of data structures and algorithms for answering visibility range queries. Data structures based on recursive subdivisions of space are popular methods for answering range queries. These data structures recursively subdivide space into cells until some criterion is satis ed. Such criterion typically involve the number of data objects that intersect the cell, but in the case of approximations may also involve the size of the cell, measured by its diameter or volume [6]. The problems of determining the size of a cell and whether a cell represents a nonempty region of space are implicit (and usually trivial) in the design of algorithms for building these data structures. In the case of directed lines in 3-space, there is a natural way of subdividing space. Given two directed lines, we may assign them an orientation, which is either positive, zero, or negative according to the direction of one line relative to a viewer looking along the direction of the other. Thus