Geometric Inversion

By Xah Lee. Date:


See: SpecialPlaneCurves_dir/Inversion_dir/inversion.html

Circle Maps to Circles

Under inversion, circles maps into circles, and lines maps into circles passing the origin.

Angle Preseservation with Reversed Sense

Under inversion, angles are unchanged.

Invariant Circles

Any point on the inversion circle maps to itself. And, any circle orthogonal to the inversional circle, will map to itself.

Considered as Circular Reflection

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.

Geometric Inversion in 3D

The inversion of a circle can be extended to 3-dimensional space. Instead of a circle, we use a given sphere. Any point inside the sphere will be mapped outside, and vice versa.