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Generalized Curvatures / by Jean-Marie Morvan
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Geometry and Computing ; 2 | SpringerLink BücherPublisher: Berlin ; Heidelberg : Springer, 2008Description: Online-Ressource (digital)ISBN:- 9783540737926
- 518
- 516.36 23
- 516.3/62
- 510
- QA71-90
- QA639.5
Contents:
Summary: Motivations -- Motivation: Curves -- Motivation: Surfaces -- Background: Metrics and Measures -- Distance and Projection -- Elements of Measure Theory -- Background: Polyhedra and Convex Subsets -- Polyhedra -- Convex Subsets -- Background: Classical Tools in Differential Geometry -- Differential Forms and Densities on EN -- Measures on Manifolds -- Background on Riemannian Geometry -- Riemannian Submanifolds -- Currents -- On Volume -- Approximation of the Volume -- Approximation of the Length of Curves -- Approximation of the Area of Surfaces -- The Steiner Formula -- The Steiner Formula for Convex Subsets -- Tubes Formula -- Subsets of Positive Reach -- The Theory of Normal Cycles -- Invariant Forms -- The Normal Cycle -- Curvature Measures of Geometric Sets -- Second Fundamental Measure -- Applications to Curves and Surfaces -- Curvature Measures in E2 -- Curvature Measures in E3 -- Approximation of the Curvature of Curves -- Approximation of the Curvatures of Surfaces -- On Restricted Delaunay Triangulations.Summary: 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.PPN: PPN: 164707990XPackage identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Introduction; Motivation: Curves; Motivation: Surfaces; Distance and Projection; Elements of Measure Theory; Polyhedra; Convex Subsets; Differential Forms and Densities on E N; Measures on Manifolds; Background on Riemannian Geometry; Riemannian Submanifolds; Currents; Approximation of the Volume; Approximation of the Length of Curves; Approximation of the Area of Surfaces; The Steiner Formula for Convex Subsets; Tubes Formula; Subsets of Positive Reach; Invariant Forms; The Normal Cycle; Curvature Measures of Geometric Sets; Second Fundamental Measure; Curvature Measures in E2
Curvature Measures in E3Approximation of the Curvature of Curves; Approximation of the Curvatures of Surfaces; On Restricted Delaunay Triangulations
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