Work

A Bicycle Correspondence on Polygons and its Continuous Limit

Year: 2025

Type: article

Abstract: We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of a completely integrable transformation on closed plane curves, known as the bicycle correspondence, that is a geometric realization of the Bäcklund transformation of the planar filament equation. For odd-gons, we construct a symplectic form on the quotient space by parallel translations and show that the transformations are symplectic, and the vector field is Hamiltonian. In the case of triangles, we prove complete integrability of the respective vector field and provide some evidence for the conjecture that the transformations are integrable as well.

Source: Journal of Experimental Mathematics

DOI: 10.56994/jxm.001.002.004

Authors: Maxim Arnold, Lael Edwards-Costa, Serge Tabachnikov

Institutions: The University of Texas at Dallas, Pennsylvania State University

Language: English

Topic: Advanced Differential Geometry Research

Subfield: Astronomy and Astrophysics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Funder: National Science Foundation

Awards: Topics in Kinematics and Geometrical Optics: Tire Track Geometry and Billiard Models, Topics in Geometrical Optics and Kinematics: Billiard Models and Tire Track Geometry