On Galilean Connections and the First Jet Bundle
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Jet Fitting 3: a Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces Via Polynomial Fitting Frédéric Cazals, Marc Pouget
Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting Frédéric Cazals, Marc Pouget To cite this version: Frédéric Cazals, Marc Pouget. Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2008, 35 (3). inria-00329731 HAL Id: inria-00329731 https://hal.inria.fr/inria-00329731 Submitted on 2 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting FRED´ ERIC´ CAZALS INRIA Sophia-Antipolis, France. and MARC POUGET INRIA Nancy Gand Est - LORIA, France. 3 Surfaces of R are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpo- lation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties. -
On Galilean Connections and the First Jet Bundle
ON GALILEAN CONNECTIONS AND THE FIRST JET BUNDLE JAMES D.E. GRANT AND BRADLEY C. LACKEY Abstract. We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations { sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces \laboratory" coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse { the \fundamental theorem" { that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. 1. Introduction The geometry of a system of ordinary differential equations has had a distinguished history, dating back even to Lie [10]. Historically, there have been three main branches of this theory, depending on the class of allowable transformations considered. The most studied has been differ- ential equations under contact transformation; see x7.1 of Doubrov, Komrakov, and Morimoto [5], for the construction of Cartan connections under this class of transformation. Another classical study has been differential equations under point-transformations. (See, for instance, Tresse [13]). By B¨acklund's Theorem, this is novel only for a single second-order dif- ferential equation in one independent variable. The construction of the Cartan connection for this form of geometry was due to Cartan himself [2]. See, for instance, x2 of Kamran, Lamb and Shadwick [6], for a modern \equivalence method" treatment of this case. -