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Linear variational principle for Riemann mappings and discrete conformality.

ABSTRACT: We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: It is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in [Formula: see text], even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the discrete Riemann mappings between each Lipschitz domain and the triangle.

SUBMITTER: Dym N

PROVIDER: S-EPMC6338830 | biostudies-other | 2019 Jan

REPOSITORIES: biostudies-other

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Linear variational principle for Riemann mappings and discrete conformality.

Dym Nadav N Slutsky Raz R Lipman Yaron Y

Proceedings of the National Academy of Sciences of the United States of America 20181228 3

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: It is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in [Formula: see text], eve ...[more]

PMID: 30593564

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