I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one place. I want to treat the $x$ and $y$ coordinates identically.
So, I have a set of points, $(x_1, y_1), ..., (x_n, y_n)$ where it is NOT necessarily the case that $x_1 < x_2 < ... < x_n$ as is the usual condition for a spline, but I'd like to use a spline-like approximation.
I'm sure this must be a problem that has been thoroughly addressed in the literature, but I can't figure out what to search for. If someone could at least tell me some keywords to search for, I would appreciate it.
Edit: After doing some more thinking and hunting around, I believe the key is going to be to use separate splines to represent the $x$ and $y$ coordinates in terms of some other parameter, $t$, i.e., as a set of parametric equations. So, for example, if I had four points, I could represent $x$ and $y$ as cubic polynomials:
$x = at^3 + bt^2 + ct + d$ and $y = et^3 + ft^2 + gt + h$. Since $t$ is an arbitrary parameter, I could let it take on integer values, e.g., $t = 0,1,2,3$ in order to make solving the system easier.
The problem is that $t$ isn't really a natural parameter. I would prefer something like arc length so that I could calculate and presumably minimize the bending energy. Somehow, I will have to incorporate that into my calculation at a later stage.