**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 + (R^{2} / |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):**

R^{2} / |z - w |^{2}.

Multiple circle inversions can strongly expand or shrink pixels.