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Geometry and Computing #2: Generalized Curvaturesby Jean Marie Morvan
Synopses & ReviewsPublisher Comments:The intent of this book is to set the modern foundations of the theory of generalized curvature measures. This subject has a long history, beginning with J. Steiner (1850), H. Weyl (1939), H. Federer (1959), P. Wintgen (1982), and continues today with young and brilliant mathematicians. In the last decades, a renewal of interest in mathematics as well as computer science has arisen (finding new applications in computer graphics, medical imaging, computational geometry, visualization ...). Following a historical and didactic approach, the book introduces the mathematical background of the subject, beginning with curves and surfaces, going on with convex subsets, smooth submanifolds, subsets of positive reach, polyhedra and triangulations, and ending with surface reconstruction. We focus on the theory of normal cycle, which allows to compute and approximate curvature measures of a large class of smooth or discrete objects of the Euclidean space. We give explicit computations when the object is a 2 or 3 dimensional polyhedron. This book can serve as a textbook to any mathematician or computer scientist, engineer or researcher who is interested in the theory of curvature measures.
Table of ContentsContents Motivations 1 Motivation Curves 1.1 The length of a curve 1.2 The curvature of a curve 1.3 The Gauss map of a curve 1.4 Curves in E2 2 Motivation Surfaces 2.1 The area of a surface 2.2 The pointwise Gauss curvature 2.3 The Gauss map of a surface 2.4 The global Gauss curvature 2.5 ... and the volume... 3 Distance and Projection 3.1 The distance function 3.2 The projection map 3.3 The reach of a subset 3.4 The Voronoi diagrams 3.5 The medial axis of a subset 4 Elements of Measure Theory 4.1 Outer measures and measures 4.2 Measurable functions and their integrals 4.3 The standard Lebesgue measure on EN 4.4 Hausdorff measures 4.5 Area and coarea formula 4.6 Radon measures 4.7 Convergence of measures 5 Polyhedra 5.1 Definitions and properties of polyhedra 5.2 Euler characteristic 5.3 Gauss curvature of a polyhedron 6 Convex Subsets 6.1 Convex subsets 6.2 Differential properties of the boundary 6.3 The volume of the boundary of a convex body 6.4 The transversal integral and the Hadwiger theorem 7 Differential Forms and Densities on EN 7.1 Differential forms and their integrals 7.2 Densities 8 Measures on Manifolds 8.1 Integration of differential forms 8.2 Density and measure on a manifold 8.3 The Fubini theorem on a fiber bundle 9 Background on Riemannian Geometry 9.1 Riemannian metric and LeviCivita connexion 9.2 Properties of the curvature tensor 9.3 Connexion forms and curvature forms 9.4 The volume form 9.5 The GaussBonnet theorem 9.6 Spheres and balls 9.7 The Grassmann manifolds 10 Riemannian Submanifolds 10.1 Some generalities on (smooth) submanifolds 10.2Thevolumeofasubmanifold 10.3 Hypersurfaces in EN 10.4 Submanifolds in EN of any codimension 10.5TheGaussmapofasubmanifold..... 140 11 Currents 11.1 Basic definitions and properties on currents 11.2 Rectifiable currents 11.3Three theorems 12 Approximation of the Volume 12.1 Thegeneralframework 12.2 A general evaluation theorem for the volume 12.3 An approximation result 12.4 Aconvergence theorem for the volume 13 Approximation of the Length of Curves 13.1 A general approximation result 13.2 An approximation by a polygonal line 14 Approximation of the Area of Surfaces 14.1 A general approximation of the area 14.2 Triangulations 14.3 Relative height of a triangulation inscribed in a surface 14.4 A bound on the deviation angle 14.5 Approximation of the area of a smooth surface by the area of a triangulation 15 The Steiner Formula for Convex Subsets 15.1 The Steiner formula for convex bodies (1840) 15.2 Examples:segments,discsandballs 15.3 Convex bodies in EN whose boundary is a polyhedron 15.4 Convex bodies with smooth boundary 15.5 Evaluation of the Quermassintegrale by means of transversal integrals 15.6 Continuity of the k 15.7 Anadditivity formula 16 Tubes Formula 16.1 The LipschitzKillingcurvatures 16.2 The tubes formulaofH.Weyl(1939) 16.3 The Eule rcharacteristic 16.4 Partial continuity of the k 16.5 Transversal integrals 16.6 On the differentiability of the immersions 17 Subsets of Positive Reach 17.1 Subsets of positive reach (H. Federer, 1958) 17.2 The Steiner formula 17.3 Curvature measures 17.4 The Euler characteristic 17.5 The problem of continuity of the k 17.6 Thetransversalintegralses 18 Invariant Forms 18.1 Invariant forms on EN Ã— EN 18.2 Invariant differential forms on EN Ã— SN1 18.3 Examplesinlow dimensions 19 The Normal Cycle 19.1 The notion of a normal cycle 19.2 Existence and uniqueness of the normal cycle 19.3 A convergence theorem 19.4 Approximation of normal cycles 20 Curvature Measures of Geometric Sets 20.1 Definition of curvatures 20.2 Continuity of the Mk 20.3 Curvature measures of geometric sets 20.4 Convergence and approximation theorems 21 Second Fundamental Measure 21.1 A vector valued invariant form 21.2 Second fundamental measure associated to a geometric set 21.3 The case of a smooth hypersurface 21.4 The case of a polyhedron 21.5 Convergence and approximation 21.6 An example of application 22 Curvature Measures in E2 22.1 Invariant forms of E2 Ã— S1 22.2 Bounded domains in E2 22.3 Plane curves 22.4 The length of plane curves 22.5 The curvature of plane curves 23 Curvature Measures in E3 23.1 Invariant forms of E3 Ã— S2 23.2 Space curves and polygons 23.3 Surfaces and bounded domains in E3 23.4 Second fundamental measure for surfaces 24 Approximation of the Curvature of Curves 24.1 Curves in E2 24.2 Curves in E3 25 Approximation of the Curvatures of Surfaces 25.1 The general approximation result 25.2 Approximation by a triangulation 26 On Restricted Delaunay Triangulations 26.1 Delaunay triangulation 26.2 Approximation using a Delaunay triangulation
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