2000/10/31 by Gregory Leibon, Leibon, Gregory
Computer Science · Mathematics · #52C26 #53A30 #60D05 #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR) #math.DG #math.PR #msc:52C26 #msc:53A30 #msc:60D05
paper · pdf · doi:10.48550/arxiv.math/0010316
openalex publication_date 2000/10/31 · arxiv created 2000/11/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately related to hyperbolic volume. With the use of random Delaunay triangulations we then average this objective function to construct an objective function on the metrics conformal to a fixed one. Finally using this averaged objective function we may reprove the uniformization theorem in two dimensions.