On the Complexity of Bicycle Unitracks

This paper concerns the geometry of bicycle tracks. We model bicycle as an oriented segment of a fixed length that is moving in the Euclidean plane so that the trajectory of the rear point is tangent to the segment at all times. The trajectories of front and back points of the segment are called bicycle tracks, and one asks if it is possible that the front track is contained in the rear track (other than when they are straight lines). Such curves are called unitracks or unicycle tracks. In 2002 D. Finn proposed a construction of unitracks that are obtained as a union of a sequence of curves. Numerical evidence suggested that these curves behave expansively and that various numerical characteristics of the curves grow quickly in the sequence. In this paper we prove that the curves that form a unitrack in Finn's construction cannot remain graphs of functions, unless they are straight lines. We conclude that the horizontal amplitude of the curves has a linear growth rate between 1 and 2.