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Generalized Curvatures / by Jean-Marie Morvan
(Geometry and Computing. ISSN:18666809 ; 2)
Edition | 1st ed. 2008. |
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Publisher | (Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer) |
Year | 2008 |
Language | English |
Size | XI, 266 p. 107 illus., 36 illus. in color : online resource |
Authors | *Morvan, Jean-Marie author SpringerLink (Online service) |
Subjects | LCSH:Geometry, Differential LCSH:Mathematics -- Data processing All Subject Search LCSH:Image processing -- Digital techniques All Subject Search LCSH:Computer vision FREE:Differential Geometry FREE:Computational Mathematics and Numerical Analysis FREE:Computer Imaging, Vision, Pattern Recognition and Graphics |
Notes | 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 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 HTTP:URL=https://doi.org/10.1007/978-3-540-73792-6 |
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E-Book | Location | Media type | Volume | Call No. | Status | Reserve | Comments | ISBN | Printed | Restriction | Designated Book | Barcode No. |
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E-Book | オンライン | 電子ブック |
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Springer eBooks | 9783540737926 |
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電子リソース |
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EB00229691 |
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