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Mathematics - Geometry & Topology | Generalized Curvatures

Generalized Curvatures

Series: Geometry and Computing, Vol. 2

Morvan, Jean-Marie

2008

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  • First coherent and complete account of this subject in book form
The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E ,), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E , endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the convex hull ofS are quantities geometric with respect toG . But the distance from the origin O to the closest point ofS is not, 0 since it is not invariant under translations ofS. It is important to point out that the property of being geometric depends on the chosen group. For instance, ifG is the 1 N group of projective transformations of E , then the property ofS being a circle is geometric forG but not forG , while the property of being a conic or a straight 0 1 line is geometric for bothG andG . This point of view may be generalized to any 0 1 subsetS of any vector space E endowed with a groupG acting on it.

Content Level » Research

Keywords » Gaussian curvature - Riemannian geometry - Riemannian manifold - computational geometry - computer graphics - curvature - curvature measure - differential geometry - discrete geometry - manifold - submanifold - triangulation - visualization

Related subjects » Computational Science & Engineering - Geometry & Topology - Image Processing

Table of contents 

Contents 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 co-area 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 Levi-Civita connexion 9.2 Properties of the curvature tensor 9.3 Connexion forms and curvature forms 9.4 The volume form 9.5 The Gauss-Bonnet 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 Lipschitz-Killingcurvatures 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 × EN 18.2 Invariant differential forms on EN × SN-1 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

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