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Curvature measures, normal cycles and asymptotic cones
Xiang Sun1; Jean-Marie Morvan2
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, Volume 3 (2013) no. 1, pp. 3-10.

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 (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].

Published online:
DOI: 10.5802/acirm.50
Classification: 00X99
Keywords: curvature measure, shape operator, surfaces, normal cycle, 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 
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Xiang Sun; Jean-Marie Morvan. Curvature measures, normal cycles and asymptotic cones. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 3-10. doi : 10.5802/acirm.50. https://acirm.centre-mersenne.org/articles/10.5802/acirm.50/

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[2] David Cohen-Steiner; Jean-Marie Morvan 4 Differential Geometry on Discrete Surfaces, Effective computational geometry for curves and surfaces (2006) | Article | Zbl: 1116.65024

[3] David Cohen-Steiner; Jean-Marie Morvan Second fundamental measure of geometric sets and local approximation of curvatures, Journal of Differential Geometry, Volume 74 (2006) no. 3, pp. 363-394 | MR: 2269782 | Zbl: 1107.49029

[4] Joseph HG Fu Monge-Ampère Functions 1, Indiana Univ. Math. J., Volume 38 (1989), pp. 745-771

[5] Joseph HG Fu Convergence of curvatures in secant approximations, Journal of Differential Geometry, Volume 37 (1993) no. 1, pp. 177-190 | MR: 1198604 | Zbl: 0794.53044

[6] Jean-Marie Morvan Generalized curvatures, 2, Springer, 2008 | MR: 2428231 | Zbl: 1149.53001

[7] Peter Wintgen Normal cycle and integral curvature for polyhedra in Riemannian manifolds, Differential Geometry. North-Holland Publishing Co., Amsterdam-New York (1982) | Zbl: 0509.53037

[8] Martina Zähle Integral and current representation of Federer’s curvature measures, Archiv der Mathematik, Volume 46 (1986) no. 6, pp. 557-567 | Article | MR: 849863 | Zbl: 0598.53058

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