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

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 π 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. ∗Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA; e-mail: [email protected]. †Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA; e-mail: [email protected] 1 ar X iv :0 80 1. 43 96 v1 [ m at h. D G ] 2 8 Ja n 20 08