  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.

Differential Geometry