Conformally symmetric triangular lattices and discrete \(\vartheta\)-conformal maps

• Published in: arXiv.org
• Main Author: Bücking, Ulrike
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 28.02.2020
• Subjects: Combinatorial analysis; Conformal mapping; Lattices; Mathematical analysis; Submerging; Triangles
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1808.08064

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.