Apparently, having mathematically defined curves that pass through a set of desired points is a thing. And (cubic) Bezier splines are popular for this. Professor Dagan (mentioned previously) sent me a link.
The article outlines a method that given a set of points you want your Bezier curve to pass through, calculate the required control points of the Bezier curve. This is similar to what I wrote here.
The difference is that my method requires the inverse of the coefficient matrix to exist, which it does. The method in that article requires the first and second derivatives of the Bezier curve to be continuous.