2007/08/18 by Ethan D. Bloch, Bloch, Ethan D.
Computer Science · Mathematics · #52B70 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.CO #math.GT #msc:52B70
paper · pdf · doi:10.48550/arxiv.0708.2506
21 pages, 1 figure
arxiv created 2007/08/18 · openalex publication_date 2007/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. We give a characterization of the angle defect for simplicial surfaces, and we show that variants of the same characterization work for two known approaches to generalizing the angle defect to arbitrary 2-dimensional simplicial complexes. Simultaneously, we give a characterization of the Euler characteristic on 2-dimensional simplicial complexes in terms of being geometrically locally determined.