Overview

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.

Full Product Details

Author:   Jean-Marie Morvan
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2008 ed.
Volume:   2
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   0.588kg
ISBN:  

9783540737919

ISBN 10:   354073791
Pages:   266
Publication Date:   13 June 2008
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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.

Reviews

From the reviews: This book is a welcome addition to the literature in differential geometry. The main aim of this book is the measure of geometric quantities describing a subset of the Euclidean space ! endowed with its standard scalar product. ! The book contains 107 figures and the bibliography contains about 89 entries. The book covers an active, interesting and fresh research area. It is very useful for researchers in differential geometry and related subjects. (Kazim Ilarslan, Zentralblatt MATH, Vol. 1149, 2008)

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