Fundamental Techniques for Geometric and Solid Modeling

We review traditional and novel paradigms for representing solids and interrogating them. The traditional paradigms reviewed are the boundary, constructive, and spatial subdivision representations. The novel representation paradigms are the B-rep index, the dimensionality paradigm, and the skeleton (medial-axis transform). The B-rep index is a polyhedral representation that integrates boundary and subdivision representation. We show how to construct it and explain some of the advantages this representation offers for operations such as point/solid and line/solid classification. We also discuss how it can account for the robustness problem. The dimensionality paradigm is a technique for representing exactly complex surfaces that satisfy prescribed constraints, and we discuss some of the algorithmic infrastructure available to manipulate and interrogate this surface representation. This paradigm generalizes both implicit and parametric representations, and can deal with surfaces that otherwise could be obtained exactly only through elimination computations of forbidding complexity. The skeleton is a solid representation originally proposed in computer vision. It is an informationally-complete solid representation, and seems to facilitate operations such as automatic mesh generation for finiteelement computations, and geometric tolerancing. We show how the skeleton relates to the cyclographic map, a concept from classical descriptive geometry, and also explain the relationship between the skeleton and the Hamilton-Jacobi equation. Finally, we review some algorithms for two-dimensional mesh generation based on the skeleton. ITo appear in Advance.!! in Conlroland Dynamic!, C. T. Leondes, ed., Academic Press.