Centroaffine Duality for Spatial Polygons

Abstract

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 Nomizu and Sasaki (Nagoya Math J 132:63–90, 1993). 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 four flattening points theorem for spatial polygons.