Estimating differential quantities using polynomial fitting of osculating jets

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...

• Type: Journal
• Year: 2005
• Where: CAGD
• Authors: Frédéric Cazals, Marc Pouget
• Keywords: CAGD 2005, Local Geometric Quantities, Smooth Manifold, Truncated Taylor Expansion