# Curvature measures, normal cycles and asymptotic cones

Xiang Sun[1]; Jean-Marie Morvan[2]

• [1] Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia
• [2] University Claude Bernard Lyon P1, France, C.N.R.S. U.M.R. 5028 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia
• Volume: 3, Issue: 1, page 3-10
• ISSN: 2105-0597

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## Abstract

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The purpose of this article is to give an overview of the theory of the normal cycle and to show how to use it to define a curvature measures on singular surfaces embedded in an (oriented) Euclidean space ${𝔼}^{3}$. In particular, we will introduce the notion of asymptotic cone associated to a Borel subset of ${𝔼}^{3}$, generalizing the asymptotic directions defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the $3$-dimensional Euclidean space ${𝔼}^{3}$. The coherence of the theory lies in a convergence theorem: If a sequence of polyhedra $\left({P}_{n}\right)$ tends (for a suitable topology) to a smooth surface $S$, then the sequence of curvature measures of $\left({P}_{n}\right)$ tends to the curvature measures of $S$. Details on the first part of these pages can be found in [6].

## How to cite

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Sun, Xiang, and Morvan, Jean-Marie. "Curvature measures, normal cycles and asymptotic cones." Actes des rencontres du CIRM 3.1 (2013): 3-10. <http://eudml.org/doc/275297>.

@article{Sun2013,
abstract = {The purpose of this article is to give an overview of the theory of the normal cycle and to show how to use it to define a curvature measures on singular surfaces embedded in an (oriented) Euclidean space $\mathbb\{E\}^3$. In particular, we will introduce the notion of asymptotic cone associated to a Borel subset of $\mathbb\{E\}^3$, generalizing the asymptotic directions defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the $3$-dimensional Euclidean space $\mathbb\{E\}^\{3\}$. The coherence of the theory lies in a convergence theorem: If a sequence of polyhedra $(P_n)$ tends (for a suitable topology) to a smooth surface $S$, then the sequence of curvature measures of $(P_n)$ tends to the curvature measures of $S$. Details on the first part of these pages can be found in [6].},
affiliation = {Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia; University Claude Bernard Lyon P1, France, C.N.R.S. U.M.R. 5028 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia},
author = {Sun, Xiang, Morvan, Jean-Marie},
journal = {Actes des rencontres du CIRM},
keywords = {curvature measure; shape operator; surfaces; normal cycle; asymptotic cones},
language = {eng},
month = {11},
number = {1},
pages = {3-10},
publisher = {CIRM},
title = {Curvature measures, normal cycles and asymptotic cones},
url = {http://eudml.org/doc/275297},
volume = {3},
year = {2013},
}

TY - JOUR
AU - Sun, Xiang
AU - Morvan, Jean-Marie
TI - Curvature measures, normal cycles and asymptotic cones
JO - Actes des rencontres du CIRM
DA - 2013/11//
PB - CIRM
VL - 3
IS - 1
SP - 3
EP - 10
AB - The purpose of this article is to give an overview of the theory of the normal cycle and to show how to use it to define a curvature measures on singular surfaces embedded in an (oriented) Euclidean space $\mathbb{E}^3$. In particular, we will introduce the notion of asymptotic cone associated to a Borel subset of $\mathbb{E}^3$, generalizing the asymptotic directions defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the $3$-dimensional Euclidean space $\mathbb{E}^{3}$. The coherence of the theory lies in a convergence theorem: If a sequence of polyhedra $(P_n)$ tends (for a suitable topology) to a smooth surface $S$, then the sequence of curvature measures of $(P_n)$ tends to the curvature measures of $S$. Details on the first part of these pages can be found in [6].
LA - eng
KW - curvature measure; shape operator; surfaces; normal cycle; asymptotic cones
UR - http://eudml.org/doc/275297
ER -

## References

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1. David Cohen-Steiner, Jean-Marie Morvan, Restricted delaunay triangulations and normal cycle, Proceedings of the nineteenth annual symposium on Computational geometry (2003), 312-321
2. David Cohen-Steiner, Jean-Marie Morvan, Effective computational geometry for curves and surfaces, (2006), Springer Zbl1107.49029
3. David Cohen-Steiner, Jean-Marie Morvan, Second fundamental measure of geometric sets and local approximation of curvatures, Journal of Differential Geometry 74 (2006), 363-394 Zbl1107.49029MR2269782
4. Joseph HG Fu, Monge-Ampère Functions 1, Indiana Univ. Math. J. 38 (1989), 745-771 Zbl0668.49010
5. Joseph HG Fu, Convergence of curvatures in secant approximations, Journal of Differential Geometry 37 (1993), 177-190 Zbl0794.53044MR1198604
6. Jean-Marie Morvan, Generalized curvatures, 2 (2008), Springer Zbl1149.53001MR2428231
7. Peter Wintgen, Normal cycle and integral curvature for polyhedra in Riemannian manifolds, Differential Geometry. North-Holland Publishing Co., Amsterdam-New York (1982) Zbl0509.53037
8. Martina Zähle, Integral and current representation of Federer’s curvature measures, Archiv der Mathematik 46 (1986), 557-567 Zbl0598.53058MR849863

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