Conformally Symmetric Triangular Lattices and Discrete ϑ-Conformal Maps

Two immersed triangulations in the plane with the same combinatorics are considered as preimage and image of a discrete immersion $F$. We compare the cross-ratios $Q$ and $q$ of corresponding pairs of adjacent triangles in the two triangulations. If for every pair the arguments of these cross-ratios (i.e., intersection angles of circumcircles) agree, $F$ is a discrete conformal map based on circle patterns. Similarly, if for every pair the absolute values of the corresponding cross-ratios $Q$ and $q$ (i.e., length cross-ratios) agree, the two triangulations are discretely conformally equivalent. We introduce a new notion, discrete $\vartheta $-conformal maps, which interpolates between these two known notions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete $\vartheta $-conformal maps are unique maximizers of a locally defined concave functional ${\mathcal{F}}_{\vartheta}$in suitable variables. Furthermore, we study conformally symmetric triangular lattices that contain examples of discrete $\vartheta $-conformal maps.

Linear variational principle for Riemann mappings and discrete conformality

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 H1, 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.