I am studying for the computer graphics chair and I have a doubt in an exercise of transformations. The exercise is as follows: The object has to reflect on a "mirrored surface" delineated by the line: ax + by = 0. The reflected object (P1 'and P2') is twice the distance of the original object (P1 and P2). The goal is to show the transformation matrix from point P to point P '. Below is an image that may help.
I can not do this exercise because I had never done a reflection on a line where I reflected "backwards" and reflected twice in the distance. Can you help me?
Even though this isn’t really a reflection, you can construct the transformation in a similar way. Decompose the arbitrary vector $\mathbf v$ into components $\mathbf v_\parallel$ and $\mathbf v_\perp$ parallel to and perpendicular to the line, respectively. The desired transformation dilates the component perpendicular to the given line by a factor of two, i.e., $\mathbf v\mapsto\mathbf v_\parallel + 2\mathbf v_\perp$. From the equation of the line you can read a vector $\mathbf n$ normal to it, and $\mathbf v_\perp$ is the orthogonal projection $\pi_{\mathbf n}\mathbf v$ of $\mathbf v$ onto $\mathbf n$, which I expect you know how to compute. Also, $\mathbf v_\parallel = \mathbf v-\mathbf v_\perp$, therefore the image of $\mathbf v$ is $\mathbf v+\mathbf\pi_{\mathbf n}\mathbf v = (I+\mathbf\pi_{\mathbf n})\mathbf v$.
Alternatively, find a rotation $R$ that takes the given line to the $x$-axis, dilate the $y$-direction by a factor of two, then rotate back, giving the matrix $$R^T\begin{bmatrix}1&0\\0&2\end{bmatrix}R$$ for this transformation.