Hyperbolic variants of Poncelet's theorem

In 1813, J. Poncelet proved his beautiful theorem in projective geometry, Poncelet's Closure Theorem, which states that: if C and D are two smooth conics in general position, and there is an n-gon inscribed in C and circumscribed around D, then for any point of C, there exists an n-gon, also inscribed in C and circumscribed around D, which has this point for one of its vertices.There are some formulae related to Poncelet's Theorem, in which introduce relations between two circles' data (their radii and the distance between their centres), when there is a bicentric n-gon between them. In Euclidean geometry, for example, we have Chapple's and Fuss's Formulae.We introduce a proof that Poncelet's Theorem holds in hyperbolic geometry. Also, we present hyperbolic Chapple's and Fuss's Formulae, and more general, we prove a Euclidean general formula, and two version of hyperbolic general formulae, which connect two circles' data, when there is an embedded bicentric n-gon between them. We formulate a conjecture that the Euclidean formulae should appear as a factor of the lowest order terms of a particular series expansion of the hyperbolic formulae.Moreover, we define a three-manifold X, constructed from n = 3 case of Poncelet's Theorem, and prove that X can be represented as the union of two disjoint solid tori, we also prove that X is Seifert fibre space