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 I: The Initial Back Tire Track

Given the relations between the front and back, we can now create an initial back-tire track segment. Recall from the description of our construction process, once we have an initial back-tire track segment, it is easy to construct an initial front-tire track segment using the relation $ \alpha(t) = \beta(t) + L T_\beta(t)$. The construction of our initial back-tire track segment follows from an interpolation scheme using conditions that can be derived from the geometry of bicycle tracks and the defining property of ambiguous tracks.

We derive the necessary and sufficient conditions on the back-tire track, by examining the relations that follow from the defining property of ambiguous bicycle tracks. Recall, the defining property of ambiguous bicycle tracks is that the front-tire track $ \alpha$ can be described as either $ \beta(t)+L T_\beta(t)$ or $ \beta(t)-L T_\beta(t)$ in terms of the back-tire track $ \beta$. This implies that for each $ t_1$ there is a $ t_2 > t_1$ such that

$\displaystyle \beta(t_1) + L\mathbf{ T}_\beta(t_1) = \beta(t_2)-L \mathbf{ T}_\beta(t_2).$ (9)
This then implies that the geometry (Frenet frames and curvatures) of the front-tire track can be computed from either description and we must obtain the same values at the corresponding $ t$-values $ t_1$ and $ t_2$. Therefore, we must have
\begin{equation*}\begin{aligned}&\left[\frac{1}{\sqrt{1 + (L\kappa_\beta)^2}} \... ...a)^2}} \mathbf{ N}_\beta \right]\Big\vert _{t=t_2} \end{aligned}\end{equation*}
and
\begin{equation*}\begin{aligned}&\left[\frac{L\frac{d\kappa_\beta}{ds_\beta}}{(\... ...t{1 + (L\kappa_\beta)^2}}\right]\Big\vert _{t=t_2}. \end{aligned}\end{equation*}

These properties will form the conditions that we need in order to generate a pair of ambiguous tire tracks. It is possible to derive additional smoothness conditions by differentiating the curvature of the front-tire track with respect to arc-length, but these are not needed in our method.

From these conditions, we can easily construct an initial back-tire track segment $ \beta:[0,1]\to \mathbb{R}^2$ which will be thrice differentiable. We first choose an isosceles triangle $ PQR$ with $ \vert PQ\vert=\vert QR\vert=L$. Then, we set $ \beta(0)=P$, $ \beta(1)=R$, and
$\displaystyle \mathbf{ T}_\beta(0) = \overrightarrow{PQ}/\vert\overrightarrow{PQ}\vert$   and$\displaystyle \quad \mathbf{ T}_\beta(1)=\overrightarrow{QR}/\vert\overrightarrow{QR}\vert.$
This ensures that

$\displaystyle Q=\beta(0) + L \mathbf{ T}_\beta(0) = \beta(1) - L \mathbf{ T}_\beta(1). $
Next, we choose a tangent vector $ \mathbf{T}_\alpha$ for the front-tire track at $ Q$ and a curvature $ \kappa_\alpha$ at $ Q$. These are used to define the curvature of the $ \beta$ at $ P$ and $ R$, and the derivative of the curvature $ d\kappa_\beta/ds_\beta$ at $ P$ and $ R$. The curvature $ \kappa_\beta$ at $ P$ and $ R$ are given respectively by
$\displaystyle \kappa_\beta(0) = \frac{(\mathbf{ T}_\alpha \cdot \mathbf{ N}_\beta)} {L (\mathbf{ T}_\alpha \cdot \mathbf{ T}_\beta)}\Big\vert _{t=0}$   
and
$\displaystyle \quad \kappa_\beta(1) = -\frac{(\mathbf{ T}_\alpha \cdot \mathbf{... ...\beta)} {L (\mathbf{ T}_\alpha \cdot \mathbf{ T}_\beta)}\Big\vert _{t=1\vert}.$
The derivative $ d\kappa_\beta/ds_\beta$ of the curvature with respect to arc-length at $ P$ and $ R$ are then given by solving
$\displaystyle \kappa_{\alpha} = \left[L\frac{d\kappa_\beta}{ds_\beta} (\mathbf{... ...\frac{(\mathbf{ T}_\alpha \cdot \mathbf{ N}_\beta)}{L}\right] \Big\vert _{t=0}$
and
$\displaystyle \kappa_{\alpha} = \left[L\frac{d\kappa_\beta}{ds_\beta} (\mathbf{... ...\frac{(\mathbf{ T}_\alpha \cdot \mathbf{ N}_\beta)}{L}\right] \Big\vert _{t=1}$
for $ \frac{d\kappa_\beta}{ds_\beta}\vert _{t=0}$ and $ \frac{d\kappa_\beta}{ds_\beta}\vert _{t=1}$.

It is now a standard exercise in interpolation to create a curve $ \beta:[0,1]\to \mathbb{R}^2$ that has these properties; prescribed values for $ \beta$, $ T_\beta$, $ \kappa_\beta$ and $ d\kappa_\beta/ds_\beta$ when $ t=0$ and $ t=1$.