  We study closed smooth convex plane curves $\Gamma$ enjoying the following
property: a pair of points $x,y$ can traverse $\Gamma$ 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 $\Gamma$, 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.
