On Bicycle Tire Tracks Geometry, Hatchet Planimeter, Menzin's Conjecture, and Oscillation of Unicycle Tracks

Abstract

The model of a bicycle is a unit segment $AB$ that can move in the plane so that it remains tangent to the trajectory of the point $A$ (the rear wheel is fixed to the bicycle frame). The same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the final position arises in this context.

According to a theorem of R. Foote, this mapping of a circle to a circle is a Möbius transformation. We extend this result to the multidimensional setting. Möbius transformations belong to one of three types: elliptic, parabolic, and hyperbolic. We prove the century-old Menzin conjecture: if the front wheel track is an oval with area at least $\pi$, then the respective monodromy is hyperbolic. We also study bicycle motions introduced by D. Finn in which the rear wheel follows the track of the front wheel. Such a ``unicycle'' track becomes more and more oscillatory in the forward direction. We prove that it cannot be infinitely extended backward and relate the problem to the geometry of the space of forward semi-infinite equilateral linkages.