Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture

TL;DR

This paper explores the geometry of bicycle tire tracks, linking it to Prytz planimeters and complex transformations, and provides a proof for a century-old conjecture by Menzin.

Contribution

It establishes new mathematical connections between bicycle tracks, planimeters, and complex analysis, and proves Menzin's long-standing conjecture.

Findings

Relation between bicycle tracks and Prytz planimeters

Proof of Menzin's 1906 conjecture

Connections with linear fractional transformations

Abstract

Geometry of the tracks left by a bicycle is closely related with the so-called Prytz planimeter and with linear fractional transformations of the complex plane. We describe these relations, along with the history of the problem, and give a proof of a conjecture made by Menzin in 1906.