The characterization of parametric surface sections

Abstract

An important class of surface intersection problems involves the sectioning of finite parametric polynomial patches by an unbounded algebraic surface. The section curve may be represented precisely by a high order algebraic curve F( u, ν) = 0 in the parameter space of the patch. It is then desired to evaluate the curve, i.e., to identify each of its open segments and closed loops, singular features such as cusps or self-intersections, and to generate ordered sequences of points along each segment or loop. Concepts from algebraic curve theory are employed to determine a set of characteristic points for the section curve. These comprise all curve points on the parameter domain boundary, all turning points where the curve tangent is parallel to u = 0 or ν = 0, and all singular points. The characteristic points dissect the section curve into a set of monotonic branches. Each characteristic point is assigned a link multiplicity, giving the number of branches entering or leaving that point. The number of monotonic curve branches is then uniquely determined by the sum of the link multiplicities. To complete the section curve evaluation, it is necessary to trace the curve branches between characteristic points. Two methods are described to identify and generate ordered point sequences along each branch: (1) locating curve points on an isoparametric grid and employing a heuristic sorting procedure; (2) marching along branches in small steps by local power-series expansions. The identification of all characteristic points coupled with the power series curve-tracing procedure provides an essentially deterministic method for evaluating parametric surface sections. This is a substantial improvement over current heuristic numerical algorithms.