Efficient CSG Representations of Two-Dimensional Solids

Abstract

Good methods are known for converting a Constructive Solid Geometry (CSG) representation of a solid into a boundary representation (b-rep) of the solid, but not for performing the inverse conversion, b-rep→CSG, which is the subject of this paper. Important applications of b-rep→CSG conversion arise in solid modeling, image processing, and elsewhere. The problem can be divided into two tasks: (1) finding a set of halfspaces that is necessary and sufficient (but not unique) to represent a given solid, and (2) constructing an efficient CSG representation using those halfspaces. This paper solves the problem for curved planar solids, i.e., r-sets in E2, with or without holes, whose boundary is given by a collection of edges. The edges may be subsets of straight lines or convex curves (i.e., curves which intersect any line in at most two points). We prove a number of results and describe algorithms that have been fully implemented for solids bounded by line segments and circular arcs. Empirical results show that the computed CSG representations are superior to those produced by earlier algorithms, and produce superior three-dimensional CSG representations for mechanical parts defined by contour sweeping. A companion paper generalizes the results to higher dimensional solids.