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Two immersed triangulations in the plane with the same combinatorics areconsidered as preimage and image of a discrete immersion $F$. We compare thecross-ratios $Q$ and $q$ of corresponding pairs of adjacent triangles in thetwo triangulations. If for every pair the arguments of these cross-ratios (i.e.intersection angles of circumcircles) agree, $F$ is a discrete conformal mapbased on circle patterns. Similarly, if for every pair the absolute values ofthe corresponding cross-ratios $Q$ and $q$ (i.e. length cross-ratios) agree,the two triangulations are discrete conformally equivalent. We introduce a newnotion, discrete $\vartheta$-conformal maps, which interpolates between thesetwo known definitions of discrete conformality for planar triangulations. Weprove that there exists an associated variational principle. In particular,discrete $\vartheta$-conformal maps are unique minimizers of a locally definedconvex functional ${\cal F}_\vartheta$ in suitable variables. Furthermore, westudy conformally symmetric triangular lattices which contain examples ofdiscrete $\vartheta$-conformal maps.

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