We have four points $ A , B, A', B'$ in the Euclidean plane. Let $O$ be the center of the direct similitude $s: AB \mapsto A'B'$.
Now, the points $A', B, B'$ being fixed, how can I find the expression in terms of complex numbers of the plane transformation $A \mapsto O$ ?
It should be an homography but I can't find its exact analytical form... Many thanks for any suggestions.
In $\mathbb{C}$ a direct similitude is $z \mapsto h(z) = \alpha z + \beta$ with $\alpha \neq 0, 1$.Let $a, a', b, b'$ the complex affixes of $A, A', B, B'$. Let us suppose $A \neq B \iff a \neq b$. The center $O$ (affixe $o$ ) is such that $h(o)= o \iff o = \frac{\beta}{1- \alpha}$. Now $\alpha$ and $\beta$ are defined by $h(a)= a'$ and $h(b) = b'$ hence $o = \frac{a b' - a'b}{a-b - (a'-b')}$. The map $a \mapsto o$ is an homography.