Properties of inversion

Inversion is a transformation of the plane with respect to a circle (called the circle of inversion) that maps each point $P$ (not the center) to a point $P'$ such that $OP \cdot OP' = r^2$, where $O$ is the center and $r$ is the radius of the circle of inversion. The properties of inversion describe how geometric objects (like lines and circles) are transformed under inversion.

Important properties

• A point on the circle of inversion stays fixed.

• The center of inversion does not have an image (it is sent to infinity).

• A line passing through the center of inversion maps to itself.

• A line not passing through the center inverts to a circle passing through the center.

• A circle passing through the center inverts to a line not passing through the center.

• A circle not passing through the center inverts to another circle not passing through the center.

• Angles between curves are preserved (inversion is conformal).