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.

Landmark-free geometric methods in biological shape analysis

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map f between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy E sd ( f ) which we introduce. This measures the distance of f from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.