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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 8, Pages 1383–1398 (Mi zvmmf609)

Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation

V. A. Garanzha

Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

Abstract: A parameterization of a surface is specified by a one-to-one mapping of a planar domain to a domain on the surface. The available approaches, which are based on conformal, quasi-conformal, and harmonic mappings, usually yield singular parameterizations when applied to nonsmooth surfaces. A variational method is considered that makes it possible to construct quasi-isometric (bi-Lipschitz) parameterizations. Estimates of the quasi-isometry (bi-Lipschitz equivalence) constants in terms of positive and negative intrinsic curvature of the surface and in terms of the so-called “pocket depth” are discussed. Numerical calculations confirm the theoretical estimates. A method for constructing computational grids on surfaces of arbitrary connectivity is proposed. This method is based on a decomposition of the surface into a set of overlapping subdomains (chart). The size of a subdomain is chosen so that the equivalence constants for its parameterization are not large. The planar grid is mapped to the surface grid. Examples of the grids generated for complex-shaped bodies with nonsmooth surfaces are presented.

Key words: bi-Lipschitz mappings, flattening, surface grids.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:8, 1334–1349

Bibliographic databases:
UDC: 519.63

Citation: V. A. Garanzha, “Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation”, Zh. Vychisl. Mat. Mat. Fiz., 45:8 (2005), 1383–1398; Comput. Math. Math. Phys., 45:8 (2005), 1334–1349

Citation in format AMSBIB
\Bibitem{Gar05} \by V.~A.~Garanzha \paper Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 8 \pages 1383--1398 \mathnet{http://mi.mathnet.ru/zvmmf609} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2191851} \zmath{https://zbmath.org/?q=an:1087.65012} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 8 \pages 1334--1349