They key point compared to typical spline or polynomial applications is that there is no element of time.
I have a sequence of points on a plane: (x1,y1), (x2,y2), ...
x is not time, and the points can loop back over each other, make a spiral, whatever.
In my application, when the points are looked at by eye, curves can be visualized (imagined), but the points do not sample along the curves with equal spacing. Sometimes there are large gaps and sometimes the points are close together.
As a cartoon example:
Just a Powerpoint hand drawn example here, I'd like the curve to be smooth and continuous.