By John Frank Adams

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The total curvature of each spherical cap is equal to 2π − Θp and does not depend on the radius of the cap. The detailed justiﬁcation can be found in [14]. A simple computation on the number of triangles, edges and vertices within the surface gives the following discrete version of Gauss-Bonnet theorem [25]: Theorem 2. Let Σ be a closed orientable singular flat surface, and χ(Σ) be the Euler characteristic of Σ. Then (2π − Θp ) = 2πχ(Σ). p vertex of Σ 5 Discrete Mean Curvature In this section, we describe two common estimators for mean curvature, the ﬁrst one deﬁned from Laplace-Beltrami operator [19], and the second one based on a cylindrical approximation.

Soc. 60, 471–502 (1990) 15. : Multi-scale Analysis of Discrete Contours for Unsupervised Noise Detection. P. ) IWCIA 2009. LNCS, vol. 5852, pp. 187–200. Springer, Heidelberg (2009) 16. fr/MeaningfulBoxes 17. : Meaningful scales detection along digital contours for unsupervised local noise estimation. IEEE Transaction on Pattern Analysis and Machine Intelligence (accepted, to appear, 2012) 18. : Digital straightness – a review. Discrete Applied Mathematics 139(1-3), 197–230 (2004) 19. : Digital Geometry: Geometric Methods for Digital Picture Analysis.

Then (2π − Θp ) = 2πχ(Σ). p vertex of Σ 5 Discrete Mean Curvature In this section, we describe two common estimators for mean curvature, the ﬁrst one deﬁned from Laplace-Beltrami operator [19], and the second one based on a cylindrical approximation. M. Mesmoudi, L. De Floriani, and P. Magillo Mean Curvature through Discrete Laplace-Beltrami Operator →p be Let H(p) be the mean curvature of a surface at a point (a vertex) p and − n the normal unit vector at p. The Laplace-Beltrami operator K maps p to the →p .