Fairing by Finite Diierence Methods + 2 F 2 Uv + F 2 Vv Du Dv (2) Mathematical Methods for Curves and Surfaces Ii 1

We propose an eecient and exible scheme to fairly interpolate or approximate the vertices of a given triangular mesh. Instead of generating a piecewise polynomial representation, our output will be a re-ned mesh with vertices lying densely on a surface with minimum bending energy. To obtain those, we generalize the nite diierences technique to parametric meshes. The use of local parameterizations (charts) makes it possible to cast the minimization of non-linear geometric functionals into solving a sparse linear system. EEcient multi-grid solvers can be applied which leads to fast algorithms that generate surfaces of high quality. x1. Introduction Fairing schemes which construct a surface by solving a constrained optimization problem, are traditionally based on piecewise polynomial representations 2,9,16,18]. The major diiculty in this approach is that on one hand eecient (linear) schemes are in general dependent on the speciic parame-terization and hence fail to be proper models of the physical or geometric intent 7]. On the other hand more sophisticated non-linear optimization is computationally involved and often unstable 15]. The basic idea of the variational design approach is to measure the quality of a surface in terms of its bending energy. The most common functional is the total curvature (1) which approximates the bending energy of a thin plate. However, since the principal curvatures and the area element depend non-linearly on the surface S, this functional is diicult to minimize. For practical fairing schemes, the total curvature is therefore replaced by the so-called thin-plate functional ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.