Parametric curve approximations for surface intersections

Abstract

A new algorithm is presented for creating a piecewise rational parametric approximation for general bivariate algebraic curves. This problem has been directly linked to the key obstacle which currently restricts the geometric coverage of solid geometric modelers, that of representing the intersection curve between two free-form surfaces. Even though numerical schemes have evolved to generate representations for these curves, there have remained critical deficiencies in the mathematical understanding and completeness of the solutions which severly limit modeling with high degree surfaces. The algorithm proposed in this thesis couples contemporary subdivision methods with techniques and insights derived from classical algebraic and projective geometry to generate the necessary piecewise approximate representation. In addition to solid geometric modeling, the piecewise C\sp0 continuous approximation is also applicable to problems in surface or scattered data contouring and in representing offset curves needed for numerical control tool path generation