linear or bilinear interpolation

Question

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to interpolate any unknown $(x,y)$ ? Do i need a weight for each point to find the points ?

Answer 1

You can define a bilinear surface from 4 points as

$S(u,v)=$$\begin{pmatrix} 1-u & u\end{pmatrix}$ $\begin{bmatrix} P_{00} & P_{01} \\ P_{10} & P_{11} \end{bmatrix}$ $\begin{Bmatrix} 1-v \\ v\end{Bmatrix}$

where $0 \le u \le 1, 0 \le v \le 1$, $P_{00}=(x_1,y_1)$, $P_{10}=(x_2,y_2)$,$P_{01}=(x_3,y_3)$ and $P_{11}=(x_4,y_4)$.