Symmetric Points

Two points $z$ and $z^S\in \Bbb{C}^*$ are symmetric with respect to a Circle or straight Line $L$ if all Circles and straight Lines passing through $z$ and $z^S$ are orthogonal to $L$. Möbius Transformations preserve symmetry. Let a straight line be given by a point $z_0$ and a unit Vector $e^{i\theta}$, then

$$z^S = e^{2i\theta} (z-z_0)^* + z_0.$$

Let a Circle be given by center $z_0$ and Radius $r$, then

$$z^S = z_0 + {r^2\over (z-z_0)^*}.$$

See also Möbius Transformation