# 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

Actes des rencontres du CIRM (2013)

- Volume: 3, Issue: 1, page 3-10
- ISSN: 2105-0597

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topSun, 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

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

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