How to determine the coordinate of roller coaster's wheels?

Question

I want to create an animation about roller coaster. For a simple track, for example, a circle, I can determine the position of the center of its wheel easily.

However, for any parametric curve, I have no idea to determine the coordinates. Let's take a general case as follows.

Given the parametric equation of a roller coaster's tracks as follows,

\begin{align} x &= f(t)\\ y &= g(t) \end{align}

where both are function of time t (for example).

A and B are the centers of the wheels with radius R. The distance between two wheels are kept constant d.

How can I determine the coordinate of A and B explicitly?

Answer 1

I would think that the distance from the track to the center of each wheel would remain constant and the function that describes the track would stay the same function, so assuming that the point on the function would be some thing like if f(t) = g(x,y) then f(t) + c = g(x,y) + c, where c is the radius of your wheel, would hold.

Answer 2

Provided the parametric curve $\vec{\gamma}(t) = (x(t), y(t))$ has a curvature smaller than $\frac{1}{R}$ (we assume the wheels always adheres to the rail), I think the curve followed by the points $A$ and $B$ is given by the $\vec{\beta}(t) = \vec{\gamma}(t) + R . \vec{N}(t)$ where $\vec{N}(t)$ is the unit normal vector.

Now let's say $A$ is at the point $\vec{\beta}(t_0)$, then $B$ is at a point $\vec{\beta}(t_1)$ with $t_1 > t_0$ such that $|\|\vec{\beta}(t_0)-\vec{\beta}(t_1)|\| = d$ (provided your track is reasonnable, and $d$ is small enough, there should be only one solution to this equation in $t_1$).