Surface design with minimum energy networks

Abstract

We describe a method for free form surface design that uses a mesh formed by networks of curves of minimum energy to define surfaces satisfying position, normal, and curvature constraints. The work addresses difficult problems arising in solid modeling, specifically at difficult blending problems, e.g. a blend at a high order vertex with simultaneous convex and concave blends incident to it. The techniques that we describe here may also be applied to general free form surface design. Previous methods for scattered data interpolation use linearized strain energy to produce a G1 network of curves which are subsequently blended. In this work we globally minimize one of two nonlinear curvature functionals, K(s)2C/J (strain energy) or f K'(s)2ds (curvature variation) ta produce a G2 network of curves. The two functionals are compared and both are shown to result in a high degree of fairness. CR Categories and Subject Descriptors: G.I.6 [Numerical Analysis]: Optimization-Constrained optimization; Gradient methods, 1.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-curve representations, J.6 [Computer Applications]: Computer-Aided Engineering-computer-aided design (CAD). Additional Key Words and Phrases: network minimization, minimum norm network, curvature, fairing, finite element method, geometric continuity, MEC, minimal energy curves, minimal variation curves, MVC, nonlinear splines, numerical optimization.