The Gauss map of polyhedral vertex stars

TL;DR

This paper provides an elementary proof connecting the discrete Gaussian curvature of polyhedral vertex stars to the algebraic area of their Gauss images, revealing geometric constraints and classifications of these images.

Contribution

It introduces a new elementary proof relating discrete Gaussian curvature to the Gauss image and derives geometric limitations on the shape of the Gauss image.

Findings

The discrete Gaussian curvature equals the algebraic area of the Gauss image.

Constraints on the shape of the Gauss image are established, including convex polygons and pseudo-quadrilaterals.

The number of inflection faces relates to positive and negative components of the Gauss image.

Abstract

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbers around the Gauss image and its antipode with its critical point index. This index is closely related to the normal degree of the Gauss map. We deduce how the number of inflection faces is related to the numbers of positive and negative components in the Gauss image. The resulting formula significantly limits the possible shapes of the Gauss image of a polyhedral vertex star. For example, if a triangulated vertex star is in general position and its Gauss image has no self-intersections, then it is either a convex spherical polygon if the curvature is positive, or a spherical pseudo-quadrilateral if the curvature is negative.