Rotation and radial translation together in a group?

Question

Being a happy beginner in algebra and groups, i would like to build a group that can systematically represent (in 2 dimensions):

1. Rotation around origin.
2. Translation along radius.

I have learned that complex numbers can represent rotation with some resolution $\theta$ : $$\bf M_{R(\theta)} = \begin{bmatrix}a&-b\\b&a\end{bmatrix}, \cases{a=\cos(\theta)\\b=\sin(\theta)}$$

And then translation with a resolution $k$ : $$\bf M_{T(k)}=\begin{bmatrix}1&0\\k&1\end{bmatrix}$$

Can I combine these somehow to achieve what I want? To build a general matrix with help of these matrices and their exponents $${\bf M(\theta,k)} = f({\bf M_{T(k)},M_{R(\theta)}},e_\theta,e_k)$$

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Edit : An image to try and clarify : We want our group to take a given line to any other given line, we measure their distance to origo and their angle. The point of interest ( which i am talking about above ) will have a $\Delta$ in each of the dimensions: angle $\theta$ and radius translation $k$.

Example where the blue line is taken to the black line by concatenating the operations: $$\cases{r = r - 0\cdot 0.125\\\theta =\theta+ 2\cdot \frac{\pi}{8}}$$ The other tuples decide the numbers for lines of the other colors. We want to find a matrix representation of generators for a group which does this. The red helper circles are centered on origo and of radiuses $0.25,0.5$ to help see what the "translation" operation is supposed to do.

Answer 1

Assuming that we start with closest point at the x-axis, one way to calculate the group action on a point $(x,y) \in {\mathbb R}^2$:

$$\begin{bmatrix}1&0&0\\0&1&0\end{bmatrix}\underset{\text{Rotation generator}}{\underbrace{\begin{bmatrix}a&-b&0\\b&a&0\\0&0&0\\\end{bmatrix}}}^{\,\,m}\underset{\text{Translation generator}}{\underbrace{\begin{bmatrix}1&0&k\\0&1&0\\0&0&1\end{bmatrix}}}^n\begin{bmatrix}x\\y\\1\end{bmatrix}$$

Or in vector form with $\bf V$ having columns being points to act on and $\bf t$ being a translation vector:

$$\begin{bmatrix}\bf I_2&\bf 0\end{bmatrix}\begin{bmatrix}\bf R&\bf 0\\{\bf 0}^T&0\\\end{bmatrix}^m\begin{bmatrix}{\bf I_2}&\bf t\\\bf 0&1\end{bmatrix}^n\begin{bmatrix}\bf V\\{\bf 1}^T\end{bmatrix}$$

The lines above were generated using this recipe, so it has been verified to work.

An illustration for comparision to the fan beam scan we here instead imagine tangential tracks on a circular rail shooting parallell rays for each rotation of the gun, translation corresponds to relocating along the track, rotation is relocating along the circle: