Moving frame and integrable system of the discrete centroaffine curves in R^3

TL;DR

This paper develops a moving frame and invariants for discrete centroaffine curves in R^3, enabling the analysis of polygon stability and periodicity through centroaffine curvatures and torsions, with applications to polygon iteration and stability.

Contribution

It introduces a novel discrete centroaffine moving frame and invariants, linking them to polygon stability and integrable systems, extending classical differential geometry methods to discrete curves.

Findings

Affine invariants distinguish discrete curves in graphics.

Polygon stability analyzed via centroaffine curvatures and torsions.

Certain polygons exhibit periodic stability under iteration.

Abstract

Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimate whether a polygon flow is stable or periodically stable. In fact, using the similar method as the Frenet-Serret frame, a discrete curve can be uniquely identified by its centroaffine curvatures and torsions. In 1878, Darboux studied the problem of midpoint iteration of polygons[12]. Berlekamp et al studied this problem in detail[2]. Now, through the centroaffine curvatures and torsions, the iteration process can be clearly quantified. Exactly, we describe the whole iteration process by using centroaffine curvatures and torsions, and its periodicity could be directly exhibited. As an application, we would obtain some stable discrete space curves with changeless curvatures and torsions after multistep iteration. For the pentagram map of a polygon, the affinely regular polygons are stable. Furthermore, we find the convex hexagons with parallel and equi-length opposite sides are periodically stable, and some convex parallel and equi-length opposite sides octagons are also periodically stable. The proofs of these results are obtained using the structure equations of the discrete cnetroaffine curves and the integrable conditions of its flows.