Mathematical Models of Rigid Solid Objects

Computational models of solid objects are potentially useful in a variety of scientific and engineering fields, and in particular in the field of design and manufacturing automation for the mechanical industries. In recent years a multitude of modelling systems have been implemented both by research laboratories and commercial vendors, but little attention has been paid to the fundamental theoretical issues in geometric modelling. This has led to severe difficulties in assessing current and proposed systems, and in distinguishing essential capabilities and limitations from user conveniences and efficiency considerations. This paper seeks a sharp mathematical characterization of "rigid solids" in a manner that is suitable for studies in design and production automation. It draws heavily on established results in modern geometry and topology. Relevant results scattered throughout the mathematical literature are placed in a coherent framework and presented in a form accessible to engineers and computer scientists. A companion paper is devoted to a discussion of representational issues in the contest set forth by this paper