Start with the reverse problem. What are the coordinates of a point fixed on the wheel. I am going to use the polar notation $r$ and $\varphi$ to locate a point on the wheel based on some reference axis. Supposed the center of the wheel is $(x_c,0)$ where $R$ is the radius of the wheel and $x_c$ is a function of time (linear). The rotation of the wheel is designated by the angle $\theta$.
The location of a point on the wheel is
$$x-x_{c}=r\,\sin(\theta+\varphi)$$
$$y=r\,\cos(\theta+\varphi)$$
So now the problem is while $x_{c}$ and $\theta$ are varying what is $r$ and $\varphi$ such that $x$ and $y$ are fixed in space. Note that $\theta\,=\,\frac{x_c}{R}$ for a rolling wheel to make everything a function of the independent variable $x_c$.
With a little math you get
$$r(x_{c})=\sqrt{x_{c}^{2}+y^{2}}$$
$$\varphi(x_{c})=\arctan\left(\frac{x_{c}}{y}\right)-\frac{x_{c}}{R}$$
With the assumption(initial condition) that $x=0$ when $x_c=0$. I plot the results for various $\frac{y}{R}$ ranging from $-1\ldots0$ and moving the center of the wheel from $x_c=-R\ldots+R$ and make some cool looking $\gamma$ looking curves. They look like the ribbons people hang to bring awareness to various illnesses. If the tire is profiled such that its width varies with radius, then only part of the shape will imprint leaving only the U shaped part of it.
Below is a graph of the shapes using Asymtpote Vector Graphics.
