Poncelet Theorems

The aim of this note is to collect some more or less classical theorems of Poncelet type and to provide them with short modern proofs. Where classical geometers used elliptic functions (or angular functions), we use elliptic curves (or degenerate elliptic curves decomposing into two rational curves). In this way we unify the geometry underlying these Poncelet type statements. Our starting point is a space Poncelet theorem for two quadrics in IP 3 (Sect. 1). This seems to have been observed first by Weyr [12] p. 28, and amplified by Griffiths and Harris [5]. The classical Poncelet theorem (cf. [6]), a statement on two conics in the plane IP 2 , follows from Weyr's space Poncelet theorem, if one of the quadrics is taken to be a cone, see Sect. 1. Gerbaldi [4] gave formulas counting the number of conics in a pencil which are in Poncelet position with respect to a fixed conic in the pencil. We show that his formulas are simple consequences of the space Poncelet theorem (Sect. 3). In Sect. 5 we evaluate explicitly the space Poncelet condition for two quadrics of revolution about the same axis. We show that theorems such as Emch's theorem on circular series [3] and (a complex-projective version of) the 'zig-zag' theorem [2] can be understood by considering torsion points on elliptic curves (see Sects. 7 and 8). Further, we prove a Poncelet version of the Money-Coutts theorem (Sect. 9). Poncelet theorems 2 Although for most of the contents of this paper only the presentation is new, it seems worth–while to us to consider Poncelet type theorems from a modern geometric point of view. In this spirit the equations of modular curves given in [1] have been transformed by N. Hitchin [7] into solutions of the Painlevé–VI–equation. And E. Previato relates Poncelet theorems to integrable Hamiltonian systems and billiards [10]. Conventions. The base field always is the field C of complex numbers. If we mention circles, quadrics of revolution, or spheres, we mean the corresponding varieties over C. Acknowledgement. This research was supported by DFG contract Ba 423/7-1 and EG contract SC1–0398–C(A).