So, I am aware of how to (both iteratively and using a linear equation) compute the cubic spline of a one-variable function with $m$ control points. However, I am not sure how to do any type of spline on a two-variable function with $mn$ control points in a square grid. Is this even possible? Is there a simple algorithm? I've tried to see if the process is separable, but so far I can't seem to prove it one way or the other.
There is a nice book by Les A. Piegl and Wayne Tiller, which is intended as an introduction to NURBS, but in the first chapters also provides theoretical background for uni- and bivariate Beziers und Splines.
The NURBS Book Les A. Piegl, Wayne Tiller Springer ISBN-10: 3540615458 ISBN-13: 978-3540615453