Article

Inequalities for the curvature of curves and surfaces

Authors:

David Cohen-Steiner,

Herbert EdelsbrunnerAuthors Info & Claims

SCG '05: Proceedings of the twenty-first annual symposium on Computational geometry

Pages 272 - 277

https://doi.org/10.1145/1064092.1064134

Published: 06 June 2005 Publication History

Abstract

In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R³ in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.

References

[1]

H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5(1995), 75--91.

[2]

T. Banchoff. Critical points and curvature for embedded polyhedra. J. Diff. Geom. 1 (1967), 245--256.

[3]

L. Broecker and M. Kuppe. Integral geometry of tame sets. Preprint, Math. Dept., WestfĠlische Wilhelms-Universität, Münster, Germany, 2000.

[4]

G. Carlsson and A. Zomorodian. Computing persistent homology. In "Proc. 20th Ann. Sympos. Comput. Geom., 2004", 347--356.

Digital Library

[5]

D. Cohen-Steiner. Topics in Surface Discretization. Ph. D. thesis, Ecole Polytechnique, Palaiseau, France, 2003.

[6]

D. Cohen-Steiner, H. Edelsbrunner and J. Harer. Stability of persistence diagrams. In "Proc. 21st Ann. Sympos. Comput. Geom., 2005", to appear.

Digital Library

[7]

D. Cohen-Steiner and J.-M. Morvan. Restricted Delaunay triangulations and normal cycles. In "Proc. 19th Ann. Sympos. Comput. Geom., 2003", 312--321.

Digital Library

[8]

M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall, Upper Saddle River, New Jersey, 1976.

[9]

H. Edelsbrunner, D. Letscher and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom. 28 (2002), 511--533.

Digital Library

[10]

I. Fáry. Sur certaines inégalités géométriques. Acta Sci. Math. Szeged 12 (1950), 117--124.

[11]

J. H. G. Fu. Convergence of curvatures in secant approximations. J. Diff. Geom. 37 (1993), 177--190.

[12]

J. Milnor. Euler characteristic and finitely additive Steiner measures. In Collected Papers, volume 1: Geometry, Publish or Perish, Houston, Texas, 1994.

[13]

J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Redwood City, California, 1984.

[14]

L. Santaló. Integral Geometry and Geometric Probability. Addison-Wesley, 1976, reprinted by Cambridge Univ. Press, England, 2004.

[15]

J. Steiner. Gesammelte Werke. Prussian Academy of Sciences, 1881, reprinted by Chelsea, New-York, 1971.

[16]

S. Tabachnikov. The tale of a geometric inequality. MASS colloquium lecture, 1999.

[17]

R. van Damme and L. Alboul. Tight triangulations. In Mathematical Methods for Curves and Surfaces, eds. M. Daehlen, L. Lyche and L. L. Schumaker, Vanderbilt Univ. Press, Nashville, Tennessee, 517--526, 1995.

Cited By

Borrelli VOrgeret F(2010)Error term in pointwise approximation of the curvature of a curveComputer Aided Geometric Design10.1016/j.cagd.2010.06.00127:7(538-550)Online publication date: 1-Oct-2010
https://dl.acm.org/doi/10.1016/j.cagd.2010.06.001
Chazal FCohen-Steiner DGlisse MGuibas LOudot SHershberger JFogel E(2009)Proximity of persistence modules and their diagramsProceedings of the twenty-fifth annual symposium on Computational geometry10.1145/1542362.1542407(237-246)Online publication date: 8-Jun-2009
https://dl.acm.org/doi/10.1145/1542362.1542407
Borrelli V(2007)Courbure discrète ponctuelleSéminaire de théorie spectrale et géométrie10.5802/tsg.24525(25-39)Online publication date: 2007
https://doi.org/10.5802/tsg.245
Show More Cited By

Index Terms

Inequalities for the curvature of curves and surfaces
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Enumeration
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Curvature formulas for implicit curves and surfaces
Special issue: Geometric modelling and differential geometry

Curvature formulas for implicit curves and surfaces are derived from the classical curvature formulas in Differential Geometry for parametric curves and surfaces. These closed formulas include curvature for implicit planar curves, curvature and torsion ...
G1 continuous approximate curves on NURBS surfaces

Curves on surfaces play an important role in computer aided geometric design. In this paper, we present a parabola approximation method based on the cubic reparameterization of rational Bézier surfaces, which generates G1 continuous approximate curves ...
Lines of curvature and umbilical points for implicit surfaces

This paper develops a method to analyze and compute the lines of curvature and their differential geometry defined on implicit surfaces. With our technique, we can explicitly obtain the analytic formulae of the associated geometric attributes of an ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SCG '05: Proceedings of the twenty-first annual symposium on Computational geometry

June 2005

398 pages

ISBN:1581139918

DOI:10.1145/1064092

Program Chairs:
Joe Mitchell
SUNY Stony Brook
,
Günter Rote
Freie Universität Berlin

Copyright © 2005 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 June 2005

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Conference

SoCG05

Sponsor:

SoCG05: The 21st Annual ACM Symposium on Computational Geometry 2005

June 6 - 8, 2005

Pisa, Italy

Acceptance Rates

SCG '05 Paper Acceptance Rate 41 of 141 submissions, 29%;

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
439
Total Downloads

Downloads (Last 12 months)16
Downloads (Last 6 weeks)1

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Borrelli VOrgeret F(2010)Error term in pointwise approximation of the curvature of a curveComputer Aided Geometric Design10.1016/j.cagd.2010.06.00127:7(538-550)Online publication date: 1-Oct-2010
https://dl.acm.org/doi/10.1016/j.cagd.2010.06.001
Chazal FCohen-Steiner DGlisse MGuibas LOudot SHershberger JFogel E(2009)Proximity of persistence modules and their diagramsProceedings of the twenty-fifth annual symposium on Computational geometry10.1145/1542362.1542407(237-246)Online publication date: 8-Jun-2009
https://dl.acm.org/doi/10.1145/1542362.1542407
Borrelli V(2007)Courbure discrète ponctuelleSéminaire de théorie spectrale et géométrie10.5802/tsg.24525(25-39)Online publication date: 2007
https://doi.org/10.5802/tsg.245
Cohen-Steiner DEdelsbrunner HMorozov DAmenta NCheong O(2006)Vines and vineyards by updating persistence in linear timeProceedings of the twenty-second annual symposium on Computational geometry10.1145/1137856.1137877(119-126)Online publication date: 5-Jun-2006
https://dl.acm.org/doi/10.1145/1137856.1137877
Cohen-Steiner DEdelsbrunner HHarer JMitchell JRote G(2005)Stability of persistence diagramsProceedings of the twenty-first annual symposium on Computational geometry10.1145/1064092.1064133(263-271)Online publication date: 6-Jun-2005
https://dl.acm.org/doi/10.1145/1064092.1064133

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents