Let's say you had a point $(a,b)$ on the complex plane. If I wanted to rotate this point $\theta$ radians around point $(c,d)$ to $(a',b')$, how would I find $a'$ and $b'$?
I'm not sure on how to get a formula for this. When rotating a complex point around the origin, I know that multiplying by fractions $i$ in exponential form works, but I'm not sure if that's the same for this.
Assume that you rotate a point $z$ around a center $w$ an angle $\theta$ to obtain $z'$.
Then
$$z'=w+e^{i\theta}(z-w)$$
The background idea is translating the $z$ and the center in order the center to be the origin, doing the rotation there, and then moving back the obtained point.
Perhaps the formula $$\frac{z'-w}{z-w}=e^{i\theta}$$ is a bit clearer.
The idea is that you can consider a traslate in the origin: $$\rho= \sqrt{(a-c)^2+(b-d)^2}$$ and $$\theta'=\arg[(a-c)+i(b-d)] $$ so now it is easy to sum $\alpha=\theta+\theta'$ and you can come back: $$a'=c+\rho\sin[\alpha]$$ and $$b'=d+\rho\cos[\alpha]$$