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Any two equivalent discrete curves must have the same invariants at thecorresponding points under an affine transformation. In this paper, weconstruct the moving frame and invariants for the discrete centroaffine curves,which could be used to discriminate the same discrete curves from differentgraphics, and estimate whether a polygon flow is stable or periodically stable.In fact, using the similar method as the Frenet-Serret frame, a discrete curvecan be uniquely identified by its centroaffine curvatures and torsions. In1878, Darboux studied the problem of midpoint iteration of polygons[12].Berlekamp et al studied this problem in detail[2]. Now, through thecentroaffine curvatures and torsions, the iteration process can be clearlyquantified. Exactly, we describe the whole iteration process by usingcentroaffine curvatures and torsions, and its periodicity could be directlyexhibited. As an application, we would obtain some stable discrete space curveswith changeless curvatures and torsions after multistep iteration. For thepentagram map of a polygon, the affinely regular polygons are stable.Furthermore, we find the convex hexagons with parallel and equi-length oppositesides are periodically stable, and some convex parallel and equi-lengthopposite sides octagons are also periodically stable. The proofs of theseresults are obtained using the structure equations of the discrete cnetroaffinecurves and the integrable conditions of its flows.

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