A process for surface fairing in irregular meshes

Abstract

This paper describes a stepwise, automatic fairing process to construct a smooth surface by optimizing suitably chosen quantitative fairness measures. The input consists of given point and/or curve data, each designated to be interpolated or approximated. These data may stem from digitizing drawings or mockup models or from prior individual curve fairing. The data are arranged in an arbitrary irregular mesh topology. The irregular, n-sided mesh cells are converted by midpoint subdivision into aggregates of quadrilateral patches (Peters, 1994), for which a biquartic Bézier surface representation is chosen everywhere. The resulting C 1 surface minimizes the fairness measure, which is selected from a variety of geometrically relevant quadratic forms, including second and higher order derivative norms. This variational formulation of the fairing problem is of Quadratic Programming type and has a unique solution. Two algorithms are described, one for global, simultaneous and another for local, iterative solution of the corresponding large linear system of equations. This surface fairing technique will be illustrated by two main examples, viz., a car hood and a twisted tripod, demonstrating the performance of the fairing algorithms and the effects of the chosen fairness measures on the character of the resulting shapes.