A discrete analog of Segre's theorem on spherical curves

TL;DR

This work formulates a discrete analogue of Segre's four-vertex theorem for space curves by introducing a discrete tangent indicatrix on the sphere and a combinatorial notion of flattenings. The main result asserts that a polygon with $\ge 4$ vertices whose discrete tangent indicatrix is not contained in any closed hemisphere and is non-self-intersecting must have at least $4$ flattenings, proven through a purely combinatorial induction on spherical polygons. The paper also develops a cone-translation lifting framework and proves discrete analogues of the Tennis Ball Theorem and Möbius-type results for spherical polygons, illustrating the broader impact of the discrete approach on 3D geometry and spherical geometry alike.

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.

Figures (12)

• Figure 1: A flattening on the left, a non-flattening on the right
• Figure 2: Flattening of a polygon $P$ and the corresponding inflection of the tangent indicatrix $Q$.
• Figure 3: A spherical polygon without inflections.
• Figure 4: Three different cases
• Figure 5: Two degenerate cases

Theorems & Definitions (51)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Definition 2.3
• Theorem 2.2
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.1
• Theorem 2.3