Estimating differential quantities using polynomial fitting of osculating jets

doi:10.1016/j.cagd.2004.09.004

Computer Aided Geometric Design

Volume 22, Issue 2, February 2005, Pages 121-146

https://doi.org/10.1016/j.cagd.2004.09.004 Get rights and content

Abstract

This paper addresses the point-wise estimation of differential properties of a smooth manifold S—a curve in the plane or a surface in 3D—assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities—such as normal, curvatures, extrema of curvature.

On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation/approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of $R^{3}$ are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.

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^☆: An extended abstract of this paper is part of the Symposium on Geometry Processing (SGP), 2003. This paper provides the proofs of the SGP paper and also features an enhanced experimental section. Work partially supported by the European Project Effective Computational Geometry for Curves and Surfaces, Shared-cost RTD (FET Open) Project No IST-2000-26473.

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Estimating differential quantities using polynomial fitting of osculating jets☆

Abstract

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General lagrange and hermit interpolation in Rn with applications to the finite element method

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Restricted Delaunay triangulations and normal cycle