In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem by the author in On behavior of pairs of Teichmüller geodesic rays, 2014 . We also consider the infimum of the asymptotic distance up to choice of base points of the rays along the geodesics. We show that the infimum is repr...

An important problem in Bayesian statistics is how the choice of prior distributions influence inferences. One way to study this is to compare values of measures of distance between prior and posterior distributions for different sample sizes and parameter values. The geodesic distance between probability distributions originally proposed by C.R.Rao is one such measure. Using formulae for the g...

Here by isometric we mean that the geodesic distance between pairs of points on the non-convex polyhedron is always equal to the geodesic distance between of the corresponding pairs of points on a cube. Alternatively, it means that two surfaces can be triangulated in such a way that they now consist of congruent triangles which are glued according to the same combinatorial rules. For example, i...

Abstract. In this paper we construct a graph-based normalisation algorithm for non-linear data analysis. The principle of this algorithm is get, in average, spherical neighborhood with unit ray. In a first paragraph we show why this algorithm can be useful as a preliminary for some neural algorithms as those that need to compute geodesic distance. Then we present the algorithm, its stochastic v...

We present an efficient computational framework for isometry-invariant comparison of smooth surfaces. We formulate the Gromov-Hausdorff distance as a multidimensional scaling (MDS)-like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances ...

In this paper, an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated. Solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold may fail to be a solution for different reasons. In particular, if a geodesic possesses conjugate p...

In this paper, the problem of non-rigid shape recognition is viewed from the perspective of metric geometry, and the applicability of diffusion distances within the Gromov-Hausdorff framework is explored. While the commonly used geodesic distance exploits the shortest path between points on the surface, the diffusion distance averages all paths connecting between the points. The diffusion dista...

A new method for obtaining the optimal path to robot navigation in 2-D environments is presented in this paper. To obtain the optimal path we use mathematical morphology in binary worlds and the geodesic distance. The navigation algorithm is based on the search for a path of minimum cost by using the wave-front of the geodesic distance of the mathematical morphology. The optimal path will be th...

A polygonal domain P with n corners V and h holes is a connected polygonal region of genus h whose boundary consists of h + 1 closed chains of n total line segments. The holes and the outer boundary of P are regarded as obstacles. Then, the geodesic distance d(p, q) between any two points p, q in polygonal domain P is defined to be the (Euclidean) length of a shortest obstacle-avoiding path bet...

This paper presents a novel method for 2D and 3D shape matching that is insensitive to articulation. It uses the eccentricity transform, which is based on the computation of geodesic distances. Geodesic distances computed over a 2D or 3D shape are articulation insensitive. The eccentricity transform considers the length of the longest geodesics. Histograms of the eccentricity transform characte...