Tire track geometry: variations on a theme

We study closed smooth convex plane curves Γ enjoying the following property: a pair of points x, y can traverse Γ so that the distances between x and y along the curve and in the ambient plane do not change; such curves are called bicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went. We discuss existence and non-existence of bicycle curves, other than circles, in particular, obtain restrictions on bicycle curves in terms of the ratio of the length of the arc xy to the perimeter length of Γ, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateral n-gons P whose k-diagonals all have equal lengths. For some values of n and k we prove the rigidity result that P is a regular polygon, and for some construct flexible bicycle polygons.