Deformation of quadrilaterals and addition on elliptic curves

Abstract

The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings. In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their angles. The space of non-oriented quadrilaterals is also an elliptic curve, doubly covered by the previous one, and is described by a biquadratic relation between the diagonals. The spaces of non-oriented quadrilaterals with the side lengths $(a_1, a_2, a_3, a_4)$ and $(s-a_1, s-a_2, s-a_3, s-a_4)$ turn out to be isomorphic via identification of two quadrilaterals with the same diagonal lengths. We prove a periodicity condition for foldings, similar to Cayley's condition for the Poncelet porism. Some applications to kinematics and geometry are presented.