On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks

TL;DR

This paper explores the geometry of bicycle and unicycle tracks, extending Moebius transformation results, proving Menzin's conjecture about track shape and monodromy type, and analyzing the oscillatory behavior of unicycle tracks.

Contribution

It extends Foote's theorem to higher dimensions, proves Menzin's conjecture, and analyzes the oscillation and extendability of unicycle tracks.

Findings

Moebius transformations of bicycle tracks are classified as elliptic, parabolic, or hyperbolic.

Menzin's conjecture is proven: oval tracks with area ≥ pi induce hyperbolic monodromy.

Unicycle tracks cannot be extended infinitely backward and relate to equilateral linkage geometry.

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 point A (the rear wheel is fixed on 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 terminal one arises. According to R. Foote's theorem, this mapping of a circle to a circle is a Moebius transformation. We extend this result to multi-dimensional setting. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. We prove a 100 years old Menzin's 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 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.