Geodesic
Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1383-1398

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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. 
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     author = {V. A. Garanzha},
     title = {Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1383--1398},
     publisher = {mathdoc},
     volume = {45},
     number = {8},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a4/}
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V. A. Garanzha. Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1383-1398. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a4/