Inequalities for the curvature of curves and surfaces

In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R3 in terms of their total absolute curvatures and the Fr´echet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and computations, Computations on discrete structures; G.2.1 [Discrete Mathematics]: Combinatorics—Counting problems General Terms Algorithms, Theory Keywords Integral geometry, curvature, Fr´echet distance, persistence diagrams, bottleneck distance, approximation, stability.

• Type: Conference
• Year: 2005
• Where: COMPGEOM
• Authors: David Cohen-Steiner, Herbert Edelsbrunner
• Keywords: COMPGEOM 2005, Discrete Geometry, Fr´echet Distance, Integral Geometry, Total Mean Curvatures