Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps

Abstract

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 discrete conformally equivalent. We introduce a new notion, discrete $\vartheta$-conformal maps, which interpolates between these two known definitions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete $\vartheta$-conformal maps are unique minimizers of a locally defined convex functional ${\cal F}_\vartheta$ in suitable variables. Furthermore, we study conformally symmetric triangular lattices which contain examples of discrete $\vartheta$-conformal maps.