A family of integrable transformations of centroaffine polygons: geometrical aspects

Two polygons, (P 1 ,...,P n ) and (Q 1 ,...,Q n ) in ℝ 2 are c-related if det(P i ,P i+1 )=det(Q i ,Q i+1 ) and det(P i ,Q i )=c for all i. This relation extends to twisted polygons (polygons with monodromy), and it descends to the moduli space of SL(2,ℝ)-equivalent polygons. This relation is an equiaffine analog of the discrete bicycle correspondence studied by a number of authors. We study the geometry of this relations, present its integrals, and show that, in an appropriate sense, these relations, considered for different values of the constants c, commute. We relate this topic with the dressing chain of Veselov and Shabat. The case of small-gons is investigated in detail.