I have a problem as follows:

I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete off-axis f(x,y) points at arbitrary x and y values, but the computation is expensive and I want to minimize the number of points I have to calculate.

How many discrete off-axis points do I need to calculate (and possibly where) to uniquely (or as close to uniquely as possible) define the surface f(x,y)?

As a starting point, I am using a Taylor series expansion of f(x,y), but so far I am unable to find the cross-terms (d2f/dxdy etc).

Note that if this is possible, I would like to extend it to 3- and 4-D functions