Generalized Curvatures
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 poin
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geometry and computing
Jean-Marie Morvan
Generalized Curvatures
2
Geometry and Computing Series Editors Herbert Edelsbrunner Leif Kobbelt Konrad Polthier
Editorial Advisory Board Jean-Daniel Boissonnat Gunnar Carlsson Bernard Chazelle Xiao-Shan Gao Craig Gotsman Leo Guibas Myung-Soo Kim Takao Nishizeki Helmut Pottmann Roberto Scopigno Hans-Peter Seidel Steve Smale Peter Schr¨oder Dietrich Stoyan
Jean-Marie Morvan
Generalized Curvatures With 107 Figures
123
Jean-Marie Morvan Universite´ Claude Bernard Lyon 1 Institut Camille Jordan Baˆ timent Jean Braconnier 43 bd du 11 Novembre 1918 69622 Villeurbanne Cedex France [email protected]
On the cover, the data of Michelangelo's head are courtesy of Digital Michelangelo Project, the image of Michelangelo's head with the lines of curvatures are courtesy of the GEOMETRICA project-team from INRIA.
ISBN 978-3-540-73791-9
e-ISBN 978-3-540-73792-6
Springer Series in Geometry and Computing Library of Congress Control Number: 2008923176 Mathematics Subjects Classification (2000): 52A, 52B, 52C, 53A, 53B, 53C, 49Q15, 28A33, 28A75, 68R c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Two Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Different Possible Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Part I: Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Part II: Background – Metric and Measures . . . . . . . . . . . . . . . . . . . . . 1.5 Part III: Background – Polyhedra and Convex Subsets . . . . . . . . . . . . 1.6 Part IV: Background – Classical Tools on Differential Geometry . . . 1.7 Part V: On Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Part VI: The Steiner Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Part VII: The Theory of Normal Cycles . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Par
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