A variational method to model G1 surfaces over triangular meshes of arbitrary topology in R3

Abstract

This article presents a method for constructing a G1-smooth surface, composed of independently parametrized triangular polynomial Bézier patches, to fit scattered data points triangulated in R3 with arbitrary topology. The method includes a variational technique to optimize the shape of the surface. A systematic development of the method is given, presenting general equations provided by the theory of manifolds, explaining the heuristic assumptions made to simplify calculations, and analyzing the numerical results obtained from fitting two test configurations of scattered data points. The goal of this work is to explore an alternative G1 construction, inspired by the theory of manifolds, that is subject to fewer application constraints than approaches found in the technical literature; e.g., this approach imposes no artificial restrictions on the tangents of patch boundary curves at vertex points of a G1 surface. The constructed surface shapes fit all test data surprisingly well for a noniterative method based on polynomial patches. Categories and Subject Descriptors: G.1.1 [Numerical Analysis]: Interpolation - Interpolation formulas; G.1.2 [Numerical Analysis]: Approximation - Spline and piecewise polynomial approximation; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling -Curve, surface, solid, and object representations; J.6 [Computer Applications]: Computer-Aided Engineering - Computer-aided design (CAD).