Short cycles of Poncelet’s conics

Main results of this paper are the following:1. A closed N-gon interscribed between two conics exists if and only if a specially constructed polygon with a smaller number of sides (n) is closed. To verify the closure of this n-gon, we need to find a periodic solution of a dynamical system of order n. The proof is based on the connection of Poncelet’s curves and matrices that admit unitary bordering [4,9,10,16]. Application of this criterion makes sense when n≪N, in particular when n≈log2N (see Table 4 where n=m1). So for example we may say that a polygon with 2049 sides interscribed between two circles is closed if and only if some specially constructed 11-gon is closed.2. A closed N-gon interscribed between two confocal ellipses (the billiard case) exists if and only if an N-gon interscribed between two special nested circles is closed.