See: SpecialPlaneCurves_dir/Inversion_dir/inversion.html
Under inversion, circles maps into circles, and lines maps into circles passing the origin.
Under inversion, angles are unchanged.
Any point on the inversion circle maps to itself. And, any circle orthogonal to the inversional circle, will map to itself.
The geometric inversion can be thought of as a reflection. Any point outside the circle will be mapped inside, and vice versa. If the inversion circle's radius increases to infinity, then we have a reflection.
The inversion of a circle can be extended to 3-dimentional space. Instead of a circle, we use a given sphere. Any point inside the sphere will be mapped outside, and vice versa.