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Generalized Curvatures

  • Jean-Marie Morvan

Part of the Geometry and Computing book series (GC, volume 2)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Motivations

    1. Pages 1-10
    2. Pages 13-28
    3. Pages 29-44
  3. Background: Metrics and Measures

  4. Background: Polyhedra and Convex Subsets

    1. Pages 71-76
    2. Pages 77-88
  5. Background: Classical Tools in Differential Geometry

  6. On Volume

  7. The Steiner Formula

  8. The Theory of Normal Cycles

  9. Applications to Curves and Surfaces

  10. Back Matter
    Pages 261-266

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • Jean-Marie Morvan
    • 1
  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 169622France

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-73792-6
  • Copyright Information Springer Berlin Heidelberg 2008
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-73791-9
  • Online ISBN 978-3-540-73792-6
  • Series Print ISSN 1866-6795
  • Buy this book on publisher's site
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