Interrogation of differential geometry properties for design and manufacture

This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.