The cycloid was the solution to a standard Freshman physics problem:
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line withouth slipping. A cycloid is a specific form of trochoid and is an example of a "roulette", a curve generated by a curve rolling on another curve.
Here is the article on rolling (in the sense of classical mechanics):
The simplest case of rolling is that of rolling without slipping along a flat surface with its axis parall to the surface (or equivalently, perpendicular to the surface normal).
The trajectory of any point is a trochoid; in particular, the trajectory of any point in the object axis is a line while the trajectory of any object rim is a cycloid.
Then they discuss the moment of inertia of the wheel itself. Example, a car wheel weights 15kgs or 30lbs.
There is a parametric equation describing the curve in this case. Vector addition yields nicely $[\text{rolling}] = [\text{translation}] + [\text{rotation}] $:
$$ \left[ \begin{array}{c} x \\ y \end{array} \right] = a \left[ \begin{array}{c} \theta \\ 1 \end{array} \right] + b \left[ \begin{array}{c} \sin \theta \\ \cos \theta \end{array} \right] $$
What happens when the wheel is no longer on a flat surface? Imagine driving a car or bicycle and the road isn't perfectly flat. We might need 3D. The response we get is quite formal:
In the case where the rolling curve is a line and the generator is a point in the line, the roulette is called an "involute" of the of the fixed curve.
So in certain cases, we can identify the specific kind of "transform" of the curve. We can recover $[circle] \times [line] = [trochoid] $ or possbly $[circle] \times [line] = [cycloid]$.
Let $r(t)$ be the "rolling" curve and $f(t)$ be the "fixed" curve then:
$$ t \mapsto f(t) + (p - r(t)) \frac{f'(t)}{r'(t)} $$
This formula is nearly useless, requiring the arc-length parameterization of a curve. For a straight line, this is tautological. A street is nearly flat, so the surface is $1 + O(\phi(x,y))$. The arc-lenght or surface area might be incomputable even with the calculus or algebra.