HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation

Estimating Differential Quantities using Polynomial fitting of Osculating Jets

Frédéric Cazals 1 Marc Pouget
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This paper addresses the pointwise 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 ^3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
Document type :
Complete list of metadata

Cited literature [28 references]  Display  Hide  Download

Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Tuesday, May 23, 2006 - 6:42:41 PM
Last modification on : Friday, February 4, 2022 - 3:18:41 AM
Long-term archiving on: : Sunday, April 4, 2010 - 10:36:54 PM


  • HAL Id : inria-00071763, version 1



Frédéric Cazals, Marc Pouget. Estimating Differential Quantities using Polynomial fitting of Osculating Jets. RR-4823, INRIA. 2003. ⟨inria-00071763⟩



Record views


Files downloads