Rotation indices related to Poncelet’s closure theorem

Abstract

Abstract. Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet's closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet's closure theorem together with Cr, with ngons for any n > k. , and our paper refers to circular versions of it. Let C R , C r be two circles with radii R > r > 0 and C r lying inside C R . From any point on C R , draw a tangent to C r and extend it to C R again, using the obtained new intersection point with C R for starting with a new tangent to C r , etc.; the system of tangential segments obtained in this way inside C R is called a Poncelet transverse (or bar billiard). We say that the annulus C R C r has Poncelet's porism property if there is a starting point on C R for which a Poncelet traverse is a closed polygon. Poncelet's closure theorem (for circles) says that then the transverse will also close for any other starting point from C R . It is known that such closing polygons (with or without self-intersections) correspond to rational rotations; e.g.,

Poncelet’s Porism in Transformation Group Framework

“…There exists an integral formula connecting , and for arbitrary given for example by Jacobi, see [6,2]. Note, that simple formula involving elliptic integrals is given in [5]. …”