WELL-FORMED SET REPRESENTATIONS OF SOLIDS

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Abstract
All point Membership Classification (PMC) algorithms on solids constructed using regularized set operations require representing and computing neighborhoods of a point with respect to the represented solid. Such computations involve some of the more sensitive and difficult algorithms in solid modeling. We establish the necessary and sufficient conditions under which set-theoretic constructions are well-formed so that PMC amounts to evaluating a ternary logic expression and does not require any neighborhood computations. We demonstrate well-formedness of any monotone set expression whose primitives form a simple arrangement in Ed, and of common representations for polyhedral objects constructed using 'decreasing convex hull' algorithms. Well-formed representations exist for every solid but not for every fixed collection of primitives. We also give a necessary and sufficient test for existence of such representations. Identifying and eliminating neighborhood computations whenever possible should lead to simpler, more efficient, and robust geometric algorithms that are also more suitable for hardware implementation. Finally, well-formed set representations can be trivially translated into representations of solids by real-valued implicit functions.