Bi-Lipschitz parameterizations of nonsmooth surfaces and surface grid generation

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.