Square surface with four fixed points

Question

I'm looking for a function of two variables, $f(x, y)$, that satisfies the following constraints:

$f(0, 0) = z_1$

$f(0, 1) = z_2$

$f(1, 0) = z_3$

$f(1, 1) = z_4$

and within the unit square, it should be "as smooth as possible". I'm not sure what to google for, but I think that I'm trying to model something like a "soap film" spanning the unit square. I found this on google images:

http://aoeu.se/so/soapfilm.gif

I suspect that I could let $f$ be the average of $z_1, ..., z_4$ weighted on the distances to the corners, but A) I'm not sure this is the best solution, and B) I was hoping for a simpler formula.

Answer 1

I know two approaches :

1. Bilinear filtering
2. Minimal surface

The first is very simple but not that nice looking. The second is much more beautiful but not hat easy to implement.

The picture looks like to be about a minimal surface.