Geometric Modeling with Algebraic Surfaces

Research in geometric modeling is currently engaged in increasing the geometric coverage to allow modeling operations on arbitrary algebraic surfaces. Operations on models often include Boolean set operations (intersection, union), sweeps and convolutions, convex bull computa· tions, primitive decomposition, surface and volume mesh generatioD, calculation of surface area and volumetric properties, etc. From these arise a number of basic problems for which effective and robust solutions need to be obtained. 'We need to devise methods for unambiguous algebraic surface model representations, for converting between alternate internal algebraic curve and surface representations such as tbe implicit and the parametric, for intersecting algebraic surfaces and topologically analyzing the inherent singularities of their high degree curve components, for soning points along algebraic curves, for minimum distance and common tangent computations between algebraic curves and surfaces, for containment classifications of algebriac curve segments and algebraic surface patches, etc. Computationally efficient algorithms for all these problems necessitate combining results from algorithmic algebraic geometry, computer algebra, computational geometry and numerical approximation theory. In tbis paper we present and discuss various such algorithms and approaches for geometric models with algebraic surfaces. ·Supported in part by NSF Grant MIP 85.:H356, ARO Contract DAAG29-85-C0018 under Cornell MSI and ONR contract NOOOl·l-S8-K·0402. Invit.ed P~per at "The Mathematics of Surfaces III", O:dord University, Oxford, UK, September 19·21,1988.