1 research outputs found
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
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