2025/04/24 by Ísmail Sağlam, Ismail Saglam, Saglam, Ismail +5 · 1 citation
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications #math.GT
paper · pdf · doi:10.48550/arxiv.2504.17328
openalex publication_date 2025/04/24 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of the edges, and the other one in terms of best Lipschitz maps. We give a description of the geodesics of this metric. We show that this metric is Finsler, and give a formula for its infinitesimal Finsler structure. We then generalise this study to the case of convex Euclidean polygons in the Euclidean plane and to surfaces equipped with singular Euclidean structures with an underlying fixed triangulation. After developing some elements of the theory of completeness and completion of asymmetric metrics which is adapted to our setting, we study the completeness of the metrics we introduce in this paper. These problems and the results obtained are motivated by Thurston's work developed in his paper Minimal stretch maps between hyperbolic surfaces. We provide an analogue of Thurston's theory in a Euclidean setting.