A way to more elegantly describe [[Affine Transformations]] as a single [[Matrix|Matrix]].
>[!example] Example simplification with homogeneous coordinates
>
>$\huge\begin{align*}
\let T: \mat{x\\y} &\mapsto \mat{x-3x+3\\2x+y+1} \\
\pa{\mat{x\\y}} &= \mat{1\\2}x + \mat{-3\\1}y + \mat{3\\1} \\
&= \mat{ 1 & -3 & 3 \\ 2 & 1 & 1} \mat{x\\y\\1}
>\end{align*}$
We assume for any coordinate in $\Rn n$, you write the coordinate in $\Rn {n+1}$ with a $1$ in the new dimension.
### Notation
The notation for a homogeneous coordinate is to put a tilde above the character.
$\huge\begin{align*}
\text{Point } P &= \pa{P_{x},P_{y}} \\
\tilde P &= \mat{ P_{x}\\ P_{y} \\ 1}
\end{align*}$
$\huge\begin{align*}
\text{Vector } \vec v &= \mat{v_{x}\\ v_{y}} \\
\tilde v &= \mat{ v_{x}\\ v_{y} \\ 0}
\end{align*}$
*Vectors have a 0 in the last column because they are not affected by translations in affine transformations*
For any [[Affine Transformations|Affine Transformation]]
$\huge T\pa{\vec x} = A\vec x+ \vec b$
In homogeneous form:
$\huge
\tilde T\pa{\vec x} = \mat{ A & \cdots & \vec b \\ \vdots & \ddots &\vdots \\ 0 & 0 & 1}
$
>[!important] Visualization
>[!definition] Simple Translation in homogeneous form
> $\huge\begin{align*}
\mat{ 1 & 0 & b_{x} \\ 0 & 1 & b_{y}\\ 0 & 0 & 1}\mat{x\\y\\1}
&= \mat{x+b_x\\y+b_y\\1}
\end{align*}$
### Line in Homogeneous Formula
$\huge \tilde l: \mat{x\\y\\1} = \mat{P_x\\P_y\\1} +t\mat{v_x\\v_y\\0}$
>[!tip] Getting fixed points in Homogeneous Form
Note that using homogeneous form will give you a whole *line* of fixed points, to get the proper $\Rn2$ fixed point where $z=1$