Deformation of Quadrilaterals and Addition on Elliptic Curves
Ivan Izmestiev
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 nonoriented quadrilaterals is also an elliptic curve, doubly covered by the previous one, and is described by a biquadratic relation between diagonals.The spaces of non-oriented quadrilaterals with the side lengths (a1, a2, a3, a4) and (s-a1, s-a2, s-a3, s-a4) 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.