On bicycle tire tracks geometry, Hatchet planimeter, Menzin’s conjecture, and oscillation of unicycle tracks. (English) Zbl 1185.37146

Levi, Mark; Tabachnikov, Serge

Exp. Math. 18, No. 2, 173-186 (2009).

Several questions regarding the geometry of bicycles are treated in this very interesting paper. First, some tools of contact geometry (Legendre curves, wave fronts, Maslov index) are used to derive a differential equation for an angle function which determines one point of the geometrical model of bicycle. Secondly, the bicycle motion is extended to higher dimensions and a characterization of parabolic monodromy is provided. The century-old Menzin conjecture is proved with the classical Wirtinger inequality, and the last section deals with oscillations of unicycle tracks. The paper contains 16 figures with detailed and useful information.

Reviewer: Mircea Crâşmăreanu (Iaşi)

MSC:

37J60	Nonholonomic dynamical systems
53A17	Differential geometric aspects in kinematics
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
70F25	Nonholonomic systems related to the dynamics of a system of particles

Keywords:

Möbius transformation; contact element; isoperimetric inequality; Wirtinger inequality; unicycle track