Aristides A. G. Requicha

The volume, moments of inertia, and similar properties of solids are defined by triple (volumetric) integrals over subsets of three-dimensional Euclidean space. The automatic computation of such integral properties for geometrically complex solids is important in CAD/CAM, robotics, and other fields and raises interesting mathematical and computational problems that have received little attention from numerical analysts and computer scientists. This paper summarizes the known methods for calculating integral properties of solids that may be geometrically complex and identifies some significant gaps in our current knowledge.

This paper discusses a family of algorithms for computing the volume, moments of inertia, and other integral properties of geometrically complex solids, e.g. typical mechanical parts. The algorithms produce approximate decompositions of solids into cuboid cells whose integral properties are easy to compute. The paper focuses on versions of the algorithms which operate on solids represented by Constructive Solid Geometry (CSG), i.e., as set-theoretical combinations of primitive solid “building blocks.” Two known algorithms are summarized and a new algorithm is presented. The efficiencies and accuracies of the three algorithms are analyzed theoretically and compared experimentally. The new algorithm uses recursive subdivision to convert CSG representations of complex solids into approximate cellular decompositions based on variably sized blocks. Experimental data show that the new algorithm is efficient and has predictable accuracy. It also has other potential applications, e.g., in producing approximate octree representations of complex solids and in robot navigation.