I just learned the trick of writing a translation of a 2-dimensional real vector as a matrix multiplication in a 3-dimensional space - wikipedia explains it here.
Basically it shows: $$ \begin{bmatrix} 1 & 0 & v_x \\ 0 & 1 & v_y\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p_x \\ p_y \\ 1 \end{bmatrix} = \begin{bmatrix} p_x + v_x \\ p_y + v_y \\ 1 \end{bmatrix} $$
My question is, why adding a third constant dimension can create/is neccesary to perform a translation via matrix-multiplication? Can it be interpretet geometrically? I would love to have an intuitive picture!
Thanks!
The geometric picture is that the coordinates you're using are actually homogeneous coordinates, and the matrix represents an affine transformation on those coordinates.
Then we can identify the linear transformations with the affine transformations fixing the last coordinate, so that all the linear transformation data is in the upper left two by two block. By fixing the last coordinate of the vectors to be 1, we can identify their other two components as "regular" vector space components, and everything lines up.
The translation, being affine but not linear, lives in the margin apart from the linear piece. As you have already seen, this wiggle room and the behavior of matrix multiplication works out to reproduce both linear transformations and translations.