Tire track geometry: Variations on a theme

We study closed smooth convex plane curves Λ enjoying the following property: a pair of pointsx, y can traverse Λ so that the distances betweenx andy along the curve and in the ambient plane do not change; such curves are calledbicycle 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, we obtain restrictions on bicycle curves in terms of the ratio of the length of the arcxy to the perimeter, length of Λ, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateraln-gonsP whosek-diagonals all have equal lengths. For some values ofn andk we prove the rigidity result thatP is a regular polygon, and for some we construct flexible bicycle polygons. Partially supported by an NSF grant.