The bilinear surface is the simplest nonflat (curved) surface because it is fully defined by means of its 4 corner points. Its four boundary curves are straight lines and the coordinates of any point on this surface are derived by linear interpolation.

Let the corner points be the four distinct points $P_{00}, P_{01}, P_{10}$ and $P_{11}$

The top and bottom boundary curves one straight lines, they are

$P(u, 0)=\left(P_{10}-P_{00}\right) u+P_{00}=P_{00}(1-u)+u P_{10}$

$P(u, 1)=\left(P_{11}-P_{01}\right) u+P_{01}=P_{01}(1-u)+u P_{11}$

To generate a curve, we calculate interpolation function using parameter V. For top boundary $v=0 $ and for bottom boundary $v=1 .$ Surface is created by linear interpolating between the top and bottom boundary.

Therefore, Resulting surface equation is,

$\begin{aligned} P(u, v) &=P(u, 0)(1-v)+v P(u, 1) \\ &=(1-v)\left[(1-u) P_{00}+u P_{10}\right]+v\left[(1-u) P_{01}+u P_{11}\right] \\ &=(1-v)(1-u) P_{00}+(1-v) u P_{10}+v(1-u) P_{01}+u v P_{11}\\ & =\left[\begin{array}{ll}{(1-u)} & {u}\end{array}\right]\left[\begin{array}{cc}{P_{00}} & {P_{01}} \\ {P_{10}} & {P_{11}}\end{array}\right]\left[\begin{array}{c}{(1-v)} \\ {v}\end{array}\right]\end{aligned}$

Where, $0 \leq u \leq 1$ and

$0 \leq u \leq 1$