Izmestiev, I. (2023). Deformation of quadrilaterals and addition on elliptic curves. MOSCOW MATHEMATICAL JOURNAL, 23(2), 205–242. https://doi.org/10.17323/1609-4514-2023-23-2-205-242
E104-03 - Forschungsbereich Differentialgeometrie und geometrische Strukturen
-
Journal:
MOSCOW MATHEMATICAL JOURNAL
-
ISSN:
1609-3321
-
Date (published):
Apr-2023
-
Number of Pages:
38
-
Publisher:
INDEPENDENT UNIV MOSCOW-IUM
-
Peer reviewed:
Yes
-
Keywords:
elliptic curves; folding of quadrilaterals; porism; biquadratic equation
en
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.
