Inversion at a circle as a mirror image

Inversion at unit circle in the complex plane:
f(z)=1/z, where z=x+iy.

Add complex conjugation to get a true mirror image.
Put the center of inversion at w=u+iv.
Use a radius R for the circle.

f(z) = w + (R2 / |z - w |2) (z - w).

Very close to the circle:
same as a mirror image at the tangent line.

Similarities far away:
- The images of circles and lines are circles and lines.
- A circle that intersects at a right angle is mapped onto itself.
- The inside of such a circle is mapped onto itself.

The image of small region is a linear transform:
Mirroring, translation, rotation and isotropic scaling.

Scaling factor (Lyapunov coefficient):

R2 / |z - w |2.

Multiple circle inversions can strongly expand or shrink pixels.