Surface Construction Based on Variational Principles

Abstract

Variational principles have become quite popular in the design of surfaces. The idea is to consider a class of surfaces having more degrees of freedom than are necessary to fullll the constraints (e. g. interpolation or boundary conditions). The remaining degrees of freedom are set by minimizing a fairness functional. The choice of thèright' fairness functional is a crucial step. In this paper we describe a systematic approach for choosing an appropriate fairness functional that produces high quality surfaces and which allows the solution of the optimization problem in a reasonable amount of time. x1. Introduction In diierent areas in CAGD arises the problem of constructing smooth curves or surfaces satisfying certain constraints. Typical examples are (scattered data) interpolation and the construction of blending surfaces. In scattered data interpolation one has to construct a surface which does not oscillate too much and goes through a sampled set of points in the space. Finding a blending surface (i.e. a smooth transition surface between primary surfaces) amounts to the construction of a reasonable surface that satisses certain boundary conditions determined by the primary surfaces. The traditional approach to these problems is the following: Choose an appropriate set of functions that has as many degrees of freedom as are necessary to fullll the constraints. Determination of the degrees of freedom usually is done by solving a system of equations. Recently, a variational approach for solving these problems has got more and more attention. The idea is to start with a class of surfaces having more degrees of freedom than are strictly necessary to fullll the constraints. The remaining degrees of freedom are used to Curves and Surfaces II 1 P.

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@inproceedings{Greiner1994SurfaceCB, title={Surface Construction Based on Variational Principles}, author={G{\"{u}nther Greiner}, year={1994} }