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
Creating Ambiguous Tracks
Part II: Extending the Tracks

To describe our method for extending the back-tire track, let $ \beta:[0,1]\to \mathbb{R}^2$ be an initial back-tire track segment constructed by one of the methods in the previous section, and let $ \alpha:[0,1] \to \mathbb{R}^2$ be the front-tire track obtained by pushing the bicycle in both possible directions on the back tire track, that is

$\displaystyle \alpha(t) = \begin{cases}\beta(1+t) - L \mathbf{T}_\beta(1+t) & ... ... \beta(t) + L \mathbf{T}_\beta(t) & \text{for $0 \leq t \leq 1$}. \end{cases}$
The construction methods for $ \beta$ in the previous section ensure that $ \alpha$ described above is twice differentiable with respect to arclength. In this section, we use the differential equations describing the geometry of bicycle tracks to extend the back-tire track $ \beta$ to $ -1 \leq t \leq 0$. We will only describe the method for extending the back-tire track to $ -1 < t < 0$, as once we have the new back-tire track we can repeat this procedure to extend the track in an iterative manner to $ t < -1$. The extension for $ t > 1$ is then established by applying the same method to the initial back-tire track $ \beta_1(t) = \beta(1-t)$ that has the opposite orientation as $ \beta$.

Let $ \Theta$ be the turning angle between $ \mathbf{T}_\alpha$ and $ \mathbf{T}_\beta$. This is defined for $ 0 \leq t \leq 1$. Our construction method is based upon extending $ \Theta$ to $ -1 \leq t \leq 0$, and then using $ \Theta$ to define the curvature of the back-tire track for $ -1 \leq t \leq 0$, and the fundamental theorem of plane curves to extend the back-tire track. The turning angle $ \Theta$ is extending by solving the differential equation

$\displaystyle \kappa_\alpha = \frac{d\Theta}{ds_\alpha} + \frac{\sin(\Theta)}{L},$
using the definition of $ \alpha$ above to compute the curvature $ \kappa_\alpha$ for $ -1 \leq t \leq 0$. The existence of a solution follows from the existence and uniqueness of solutions to differential equations at least for a short time $ -\epsilon < t < 0$. However, it is not hard using comparison theorems to show that there is a solution for $ -1 \leq t \leq 0$. But, it is not possible to show that a generic solution must satisfy $ -\frac{\pi}{2} < \Theta < \frac{\pi}{2}$. In fact, one can easily construct examples of initial segments for which the solution fails to satisfy $ -\frac{\pi}{2} < \Theta < \frac{\pi}{2}$.

Rather than give a formal study of the existence of solutions satisfying $ -\frac{\pi}{2} < \Theta < \frac{\pi}{2}$, we supply a informal perturbation argument that one can extend some initial segments practically indefinitely. We first note that if $ \Theta$ is identically a constant on $ 0 \leq t \leq 1$, then $ \Theta$ extends to a constant for $ -1 \leq t \leq 0$. Thus, an initial segment of constant $ \Theta$ (an arc of a circle or a line segment) will generate a circle or a straight line (curves of constant curvature). If we start with a small perturbation of a segment of constant curvature, then it seems reasonable that with a small enough perturbation one can extend the track practically indefinitely. Of course, we will not specify what we mean by small.

Our argument relies on the use of comparison theorems to show the existence of a solution $ \Theta$ for $ -1 \leq t \leq 0$, and the continuous dependence of the solution on the parameters of the differential equation that is the curvature of the front tire $ \kappa_\alpha$ and the initial condition $ \Theta(0)$. The existence of a solution for $ -1 \leq t \leq 0$ is obtained by comparing the solution $ \Theta$ to solutions of the linear equations

$\displaystyle \frac{d\Theta}{ds} + \frac{\Theta}{L} = \kappa_\alpha$    and $\displaystyle \quad \frac{d\Theta}{ds} - \frac{\Theta}{L} = \kappa_\alpha. $
From the continuous dependence on the parameters, it follows that if $ \kappa_\alpha$ is close to a constant, then $ \Theta(t)$ is close to $ \Theta(0)$ for $ -1 \leq t \leq 0$ therefore $ -\frac{\pi}{2} < \Theta(T) < \frac{\pi}{2}$. Furthermore, $ \Theta(t)$ is close to $ \Theta(0)$ for $ -1 \leq t \leq 0$ and $ \kappa_\alpha$ is close to a constant for $ -1 \leq t \leq 0$. The iterative nature of our construction then implies that we should be able to apply the same argument practically indefinitely as long as we start with a sufficiently small perturbation of a curve of constant curvature, which can be extended indefinitely. The catch in this heuristic argument is that the technical meaning of the word close may change as the iterate along. However, if we start with a very small perturbation we should be able to extend the curve as far as we want.