Suppose I have some polar coordinate system $\mathbf{x} (r,\theta$). At location $ \mathbf{x}_A =(10,0)$ I have some point $A$. I also have point B at $\mathbf{x}_B =(20,\pi/2)$.
Now suppose I want to shift the origin of my coordinate system such that it is centered on $\mathbf{x}_A$. This new coordinate system I denote $\mathbf{x'} (r',\theta'$).
Clearly now $\mathbf{x'}_A = 0,0$. I want to determine the transformation for $\mathbf{x}_B \rightarrow \mathbf{x'}_B $ and also the general transformation so as to map any $\mathbf{x} \rightarrow \mathbf{x'} $
Any ideas?
You can use cosine rule to get $r'$ and some trigonometry to get $\theta'$
$$r' = \sqrt{r^2+10^2 - 2\cdot 10 \cdot r \cos(\theta)}\\ \tan(\theta') = \dfrac{r\sin(\theta)}{r\cos(\theta)-10}$$