>[!tip] > This entire page basically describes [[../Math/Barycentric Coordinates|Barycentric Coordinates]] but for [[Rectangle|Rectangles]] To [[Interpolate|Interpolation]] a value at position $I$ within a [[Rectangle]] $\square ABCD$ with each point having some value $\kappa_{x}$ that we are trying to interpolate between. Assuming in $\square ABCD$, $A$ is in the bottom left of the [[Cartesion Plane]] and the following points $B,\, C,\, D$ are placed in [[Counter-Clockwise]] positions around the center of the rectangle, Let $s$ be the [[Distance]] between $I$ and $\overline{AD}$. Let $t$ be the [[Distance]] between $I$ and $\overline{AB}$. $\huge \begin{align} P&=\op{lerp}\pa{A, B, s}\\ Q&=\op{lerp}\pa{D, C, s}\\ I &= \op{lerp}\pa{P, Q, t}\\ \\ P &= A +(B-A)s \\ Q &= D + (C-D)s \\ I &=P + (Q-P)t \\ &= A+(B-A)s + t\pa{ D+(C-D)s - A-(B-A)s} \\ &\cdots\\ I &= (1-s)(1-t) A + s(1-t) B + st C + (1-s)t D \\ &=\lambda A + \mu B + \nu C + \zeta D \\ &= \ba{\lambda , \mu, \nu, \zeta}_{\square ABCD} \end{align} $ $\huge \begin{align} \lambda &= (1-s)(1-t) \\ \mu &= s(1-t)\\ \nu &= st\\ \zeta &= (1-s)t\\ \end{align} $ $\huge \begin{align} \mathcal{C}_{\square ABCD \to S} &= \mat{ A & B & C & D } \end{align}$ >[!tip] > $\lambda, \mu, \nu, \zeta$ are the normalised areas of each quadrant formed by placing I within $\square ABCD$ > $\huge \lambda+\mu+\nu+\zeta = 1 $