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Moscow Mathematical Journal

Volume 23, Issue 2, April–June 2023  pp. 205–242.

Deformation of Quadrilaterals and Addition on Elliptic Curves

Authors:  Ivan Izmestiev (1)
Author institution:(1) Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria


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 $(a_1, a_2, a_3, a_4) \leftrightarrow (s-a_1, s-a_2, s-a_3, s-a_4)$ 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.

2020 Math. Subj. Class. 52C25, 33E05.

Keywords: Folding of quadrilaterals, porism, elliptic curve, biquadratic equation.

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