Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture

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Abstract

We study a simple model of bicycle motion: A bicycle is a segment of fixed length that can move in the plane so that the velocity of the rear end is always aligned with the segment. The same model describes the hatchet planimeter, a mechanical device for approximate measuring area of plane domains. 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. This circle mapping is a Möbius transformation, a remarkable fact that has various geometrical and dynamical consequences. Möbius transformations belong to one of the three types: elliptic, parabolic, and hyperbolic. We describe a proof of a 100-year-old conjecture: If the front wheel track of a unit bike is an oval with area at least π, then the respective monodromy is hyperbolic.

Original languageEnglish (US)
Pages (from-to)199-216
Number of pages18
JournalAmerican Mathematical Monthly
Volume120
Issue number3
DOIs
StatePublished - Mar 1 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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