# Centroaffine Duality for Spatial Polygons

## TL;DR

This paper explores the centroaffine geometry of spatial polygons, establishing duality relations, vertex-flattening point correspondences, and applications to flattening point theorems.

## Contribution

It introduces a new duality framework for spatial polygons and proves novel correspondences and properties within centroaffine geometry.

## Key findings

- Vertices correspond to flattening points in dual polygons
- Constant curvature polygons are dual to planar polygons
- Provides a new proof of the 4 flattening points theorem

## Abstract

In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for $(Y,V)$ and also that constant curvature polygons are dual to planar polygons. As an application, we give a new proof of a known $4$ flattening points theorem for spatial polygons.