Deformation of quadrilaterals and addition on elliptic curves

Abstract

The space of quadrilaterals with fixed side lengths is an elliptic curve, for a generic choice of lengths. Darboux used this fact to prove his porism on foldings. We study the spaces of oriented and non-oriented quadrilaterals with fixed side lengths. This is done with the help of the biquadratic relations between the tangents of the half-angles and between the squares of the diagonal lengths, respectively. The duality (a₁ , a₂ , a₃ , a₄ ) ↔ (s − a₁ , s − a₂ , s − a₃ , s − a₄ ) between quadruples of side lengths turns out to preserve the range of the diagonal lengths. In particular, the corresponding spaces of non-oriented quadrilaterals are isomorphic. We show how this is related to Ivory’s lemma. Finally, we prove a periodicity condition for foldings, similar to Cayley’s condition for the Poncelet porism.