Tire track geometry: variations on a theme
| dc.creator | Tabachnikov, Serge | |
| dc.date | 2004-05-24 | |
| dc.date.accessioned | 2026-07-07T05:08:30Z | |
| dc.date.available | 2026-07-07T05:08:30Z | |
| dc.description | We study closed smooth convex plane curves $Γ$ enjoying the following property: a pair of points $x,y$ can traverse $Γ$ so that the distances between $x$ and $y$ along the curve and in the ambient plane do not change; such curves are called {\it bicycle curves}. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went. We discuss existence and non-existence of bicycle curves, other than circles, in particular,obtain restrictions on bicycle curves in terms of the ratio of the length of the arc $xy$ to the perimeter length of $Γ$, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateral $n$-gons $P$ whose $k$-diagonals all have equal lengths. For some values of $n$ and $k$ we prove the rigidity result that $P$ is a regular polygon, and for some construct flexible bicycle polygons. | |
| dc.identifier | https://arxiv.org/abs/math/0405445 | |
| dc.identifier | http://arxiv.org/abs/math/0405445 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71291 | |
| dc.subject | Differential Geometry | |
| dc.title | Tire track geometry: variations on a theme | |
| dc.type | text |