Poncelet porism

For two smooth plane conics meeting transversely, the operation of crossing a chord of the second conic tangent to the first and choosing the other tangent through the new endpoint acts as a translation on an elliptic curve on their incidence curve. Each switch is an involution of a degree-two map from a genus one curve. One periodic orbit means that the translating point is torsion, so every orbit is periodic with the same least period. The construction extends through coincident choices at ramification points using the regular involutions.