Вычислительная математика - ИСЭМ СО РАН

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COMPUTATION OF DISCRETE CURVATURES USING CONCEPT OF
POLAR POLYHEDRA
V.A. Garanzha
Computing Center RAS, Moscow
e-mail: garan@ccas.ru
Abstract. Duality principle for computation of boundary length and area of convex domains was
introduced by ancient greek geometer and physicist Archimedes. In fact he introduced numerical
method for computation of arclength and area of a circle via sequence of inscribed and circumscribed
polygons. This idea allowed to prove convergence of numerical approximations to arclength and area
and to obtain two-sided error estimates. In this work it is shown that duality principle allows to
obtain convergent piecewise-affine approximation to spherical Gauss map when regular surface is
approximated by a sequence of pairs of locally polar polyhedra. As a result on can obtain convergent
discrete pointwise approximations to mean and Gaussian curvature and to curvature tensor as well as
convergent discrete approximations to integral curvature measures.
Key words: discrete curvature, polar polyhedra, surfaces of bounded curvature, bending energy
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