A discrete analog of Segre's theorem on spherical curves

TL;DR

This paper establishes a discrete version of Segre's four-vertex theorem for space polygons, showing that polygons with certain properties must have at least four flattening points, analogous to torsion vanishing points in smooth curves.

Contribution

It introduces a discrete analog of Segre's theorem, extending the classical smooth curve result to polygons using discrete tangent indicatrices.

Findings

Polygons with at least four vertices and non-self-intersecting discrete tangent indicatrices have at least four flattenings.

The discrete flattenings correspond to triples of vertices with specific geometric properties.

The approach bridges smooth and discrete differential geometry for space curves.

Abstract

We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at least four points at which its torsion vanishes. Our approach uses the notion of discrete tangent indicatrix of a (closed) polygon. Our theorem then states that a polygon with at least four vertices and whose discrete tangent indicatrix has no self-intersections admits at least four flattenings, i.e., triples of vertices such that the preceding and following vertices are on the same side of the plane spanned by this triple.