Interpolate a rectangular surface with given edges I need to interpolate a surface by filling a rectangular hole. The height values of the edges are given.  : \mathbb{R^2} \to \mathbb{R}$ that assigns a height value for every point of the surface.

Let's call $f_0(y)$ the values on the segment $\\{0\\}\times[0,1]$, $f_1(y)$ the values on $\\{1\\}\times [0,1]$, and similarly $g_0(x)$ and $g_1(x)$.

A continuous function interpolating them is $$ \frac{[(1-x)f_0(y) + xf_1(y)]y(1-y) + [(1-y)g_0(x) + yg_1(x)]x(1-x)}{x(1-x) + y(1-y)} $$