Exact Mathematical Statement of the Problem:

Do there exist any curves  parameterized with respect to arclength such that the positions of the front tire and back tire of a bicycle at time , respectively satisfy   and  for some positions  and  of the curve ?

 

Assumptions:

We will assume that the bicycle is ridden on a flat surface, and that the bicycle is not banked into the ground, that is the plane of each tire meet the ground in a right angle. We suspect that one can create a unicycle track without these assumptions, but the equations become much more complicated.  In particular, these assumptions imply that

(1)

 

where L is the length of the bicycle and  is the unit tangent vector to the back tire  at time t. Equation (1.1) is an important relation which allows us to derive equations for generating bicycle tracks.

 

 

Solution: For details on how it is possible to create a unicycle track with a bicycle see the paper "Can a Bicycle Create a Unicycle Track?".  The main ideas are to first create a smooth initial track by perturbing a straight line, and  then to push the bicycle forward on this track keeping the one of the tires in contact with the initial segment (meaning the tangent line of the track is the parallel to the unit tangent vector of the tire's position).   Several problems present themselves in creating a unicycle track with a bicycle. The first of which is  creating a initial segment of track that allows the above procedure to be implemented.  Once that is solved, the second problem is deriving the equations describing the relations between the front and back tires of a bicycle.  These equations are necessary to extend the initial segment to a complete unicycle track; a parametric curve defined for all values of s the arclength parameter.  An more detailed outline of the solution can be found in the details of the construction and numerical construction by the following links below (or the links on the menu)

 

Warning the links above contain mathematical equations stored as GIF files and may take a short while to load.