A family of maps and a vector field on plane polygons

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 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 evidence for the conjecture that the transformations are integrable as well.

Figures (14)

• Figure 1: Left: ${\bf Q} = q_t({\bf P}), {\bf R} = r_t({\bf P})$. Right: two curves in the bicycle correspondence.
• Figure 2: Constructing $q_t^{-1}$.
• Figure 3: The vector field $\xi$.
• Figure 4: To proof of Proposition \ref{['prop:form']}.
• Figure 5: Deducing the formula for integral $I$.

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Theorem 1