Transform point in different coordinate system (rotated and translated)

Question

I have 3 points: $A_s(x_{a_{s}}|y_{a_{s}})$, $B_s(x_{b_{s}}|y_{b_{s}})$ and $P_s(x_{p_{s}}|y_{p_{s}})$ of which i have each coordiate. (Black System)

I want to get point $P_r(x_{p_{r}}|y_{p_{r}})$ in the Red System. I know $A_r(x_{a_{R}}|y_{a_{r}})$, $B_r(x_{b_{r}}|y_{b_{r}})$

The proportions of the triangle $\bigtriangleup_{ABP}$ (angles and relative distances) don't change

I've heard of roatation and translation matrices but i don't know how to calcute it.

Answer 1

This is best solved using complex numbers. A similarity transform is simply written

$$Z=pz+q$$ and if there is no scaling (rigid transform), $|p|=1$.

By the usual formula,

$$Z=\frac{Z_b-Z_a}{z_b-z_a}(z-z_a)+Z_a$$

If your data points fulfill the isometric condition, $\left|\dfrac{Z_b-Z_a}{z_b-z_a}\right|=1$ will implicitly hold.