I was hoping someone would mention the variational properties of splines, so I'm glad to see you bring up their energy minimization. In Strang's Intro to Applied Mathematics, he briefly mentions (at the end of the section on splines) that a spline interpolation can be found by minimum principles, since it's essentially passing a beam in bending through the control points. I was curious if anyone has used that practically, eg writing a finite element spline solver?
I am unaware of any effort in that direction, but I did find a 1987 paper by Höllig [1] that seems to do a kind of converse (approximates FEM solutions using splines. I haven't read the paper, but that's my quick take). Recovering splines using a FEM might be an interesting learning exercise, since the analytic solution is known and the model seems simple. The motivating physical model is described by flat splines [2], and Carl de Boor has photographs of physical splines in action [3].
