Method for intersecting algebraic surfaces with rational polynomial patches

Abstract

The paper presents a hybrid algorithm for the computation of the intersection of an algebraic surface and a rational polynomial parametric surface patch. This algorithm is based on analytic representation of the intersection as an algebraic curve expressed in the Bernstein basis; automatic computation of the significant points of the curve using numerical techniques, subdivision and convexity properties of the Bernstein basis; partitioning of the intersection domain at these points; and tracing of the resulting monotonic intersection segments using coarse subdivision and faceting methods coupled with Newton techniques. The algorithm described in the paper treats intersections of arbitrary order algebraic surfaces with rational biquadratic and bicubic patches and introduces efficiency enhancements in the partitioning and tracing parts of the solution process. The algorithm has been tested with up to degree four algebraics and bicubic patches.