Discrete extrinsic curvatures based on polar polyhedra concept

TL;DR

This paper introduces a novel approach using polar polyhedra to discretize and approximate extrinsic curvatures of convex bodies, extending classical geometric approximation methods.

Contribution

It presents a new method for approximating extrinsic curvatures via polar polyhedra, enabling convergence to smooth curvature measures and discretization of bending energies.

Findings

Convergent point-wise approximations to mean and Gauss curvature.

Natural discretizations of bending energies.

Extension of classical approximation methods to extrinsic curvature.

Abstract

Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to scherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies.