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 Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 1, Pages 71–98 (Mi zvmmf4813)

Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra

V. A. Garanzha

119333 Moscow, ul. Vavilova, 40, Dorodnicyn Computing Center, RAS

Abstract: Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate “Measurement of circle”. The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to spherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies. Suggested approach can be applied to nonconvex surfaces and in the case of multiple dimensions.

Key words: polar polyhedra, discrete curvatures, DC surfaces (representable as a difference of convex functions), bending energy.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:1, 65–92

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UDC: 519.53
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Citation: V. A. Garanzha, “Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra”, Zh. Vychisl. Mat. Mat. Fiz., 50:1 (2010), 71–98; Comput. Math. Math. Phys., 50:1 (2010), 65–92

Citation in format AMSBIB
\Bibitem{Gar10} \by V.~A.~Garanzha \paper Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2010 \vol 50 \issue 1 \pages 71--98 \mathnet{http://mi.mathnet.ru/zvmmf4813} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2681136} \elib{https://elibrary.ru/item.asp?id=13044701} \transl \jour Comput. Math. Math. Phys. \yr 2010 \vol 50 \issue 1 \pages 65--92 \crossref{https://doi.org/10.1134/S0965542510010082} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-76649086415} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Klyachin, A. A. Shirokii, “The Delaunay triangulation for multidimensional surfaces and its approximative properties”, Russian Math. (Iz. VUZ), 56:1 (2012), 27–34
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