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

TL;DR

This paper introduces discrete $ heta$-conformal maps as an interpolation between existing notions of discrete conformality, proves a variational principle for them, and explores symmetric lattices that exemplify these maps.

Contribution

It defines discrete $ heta$-conformal maps, establishes their variational principle, and analyzes symmetric lattices as examples.

Findings

Existence of a convex functional ${ m extbf{F}}_ heta$ for these maps.

Discrete $ heta$-conformal maps are unique minimizers of this functional.

Identification of symmetric lattices that admit such 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.