# surface of the form $z(x,y)=a xy+b x+c y+d$ through four given points

I know, that for any four points, $(x_0,y_0,z_{00})$, $(x_1,y_0,z_{10})$, $(x_0,y_1,z_{01})$ and $(x_1,y_1,z_{11})$, if $x_0\ne x_1$ and $y_0\ne y_1$, there is a unique surface of the form $z(x,y)=a xy+b x+c y+d$ passing through these points.

Can someone please guide me to a resource that discusses these type of surfaces at an introductory level? I am trying to guide high school students to do some experiment in this area without me telling them what to do. An elementary introduction that starts with the equation of a plane through three points and moves towards how to modify a plane equation if a fourth point is added would be ideal. I tried to search on the internet, but without much success.

Existence and uniqueness go with this linear problem you don't seem to be mentioning: $$\left( \begin{array}{rrrr} x_0 y_0 & x_0 & y_0 & 1 \\ x_0 y_1 & x_0 & y_1 & 1 \\ x_1 y_0 & x_1 & y_0 & 1 \\ x_1 y_1 & x_1 & y_1 & 1 \end{array} \right) \left( \begin{array}{r} a \\ b \\ c \\ d \end{array} \right) = \left( \begin{array}{r} z_{00} \\ z_{01} \\ z_{10} \\ z_{11} \end{array} \right)$$
• It's a 2d equivalent of a 1d linear interpolation, a tensor product linear interpolation: $s(x,y)=(ax+b)(cy+d)$. The 4 unknown coefficients are just enough to be determined by 4 points (uniquely if the points are unisolvent). – rych Dec 4 '16 at 8:50