Which Way Did You Say
That Bicycle Went? 		David L. Finn
Rose-Hulman
Institute of Technology

Problem Statement I

Problem Statement II
Solution to Original
Problem

Outline of Construction
of Ambiguous Tracks

Animations and Examples
of Ambiguous Tracks

Geometry of Tire Tracks

Part I of Solution:
Creating an Initial
Piece of Track

Part II of Solution
Extending the Track

References
Problem Statement II:
Solution to the Original Problem
To determine which direction a bicycle went from its tire tracks, we use the relations between the positions of the front tire and the back tire. We assume for simplicity that the bicycle is ridden on a perfectly flat surface and that the plane of the tires meet the plane of the surface in a right angle. Without these assumptions, the problem becomes much much harder. With these assumptions and a couple of facts about the construction of a bicycle, we have that the position of the front tire $ \alpha(t)$ at time $ t$ is related to the position of the back tire $ \beta(t)$ at time $ t$ by
$\displaystyle \alpha(t) = \beta(t) + L\,\mathbf{ T}_\beta(t),$ (1)
where $ \mathbf{ T}_\beta(t)$ is the unit tangent vector of the back-tire track and $ L$ is a constant representing which represents the length of the bicycle.

We can now attempt to determine the direction a bicycle was travelling from its tire tracks using (1). First, we need to determine which track was created by the front tire and which was created by the back tire. One can do this by trial and error. However, it is normally possible to determine which tire is which by inspection using the simple fact that the front-tire track should vary more than the back-tire track. This means the curve that deviates more from a straight line (has a larger amplitude) should be the front-tire track, and the curve that deviates less from a straight line (has a smaller amplitude) should be the back-tire track. Once, we know which tire is which, all we need to do is draw tangent lines at a few points on the back-tire track and measure the distance between the point and tangency and the intersections with the front-tire track. The bicycle then went in the direction in which the measured distance does not change.