Consider the 1D line embedded in \(\mathbb{R}^2\) defined by the point \(\mathbf{p} = \begin{bmatrix}1 & 1\end{bmatrix}\) and the normal vector \(\mathbf{n} = \begin{bmatrix}1 & 1 \end{bmatrix}\). Notice that this is an implicit representation for the line, where a point \(\mathbf{x}\) is on the line if \((\mathbf{p} - \mathbf{x}) \cdot \mathbf{n} = 0\).
Consider applying the following 2D affine transformation \(M\) to this line by directly transforming \(\mathbf{p}\) and \(\mathbf{n}\): \[ M = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \] You’ll find that the line defined by the point \(M\mathbf{p}\) and normal vector \(M\mathbf{n}\) does not represent the set of points on the original line transformed by \(M\). What happened, and how can we fix this?
Consider an axis-aligned 2D unit square whose bottom-left corner is located at the origin. In this problem, we are dealing in linear transformations only, so there’s no need to write anything in homogeneous coordinates.