Inversive Geometry (Inversion)

Inversion in a circle: the relation OP · OP' = r^2 and how it moves lines and circles.

The idea

Inversion in a circle is a transformation of the plane that mirrors points across a circle, as reflection mirrors them across a line.

Fix a center $O$ and a radius $r$. Inversion in that circle sends each point $P$ other than $O$ to the point $P'$ on the ray from $O$ through $P$ for which

$OP \cdot OP' = r^{2}.$

The constant $r^{2}$ is called the power of the inversion. Because the product of the two distances is fixed, the map sends points near $O$ far away and distant points close in, and it exchanges the inside of the circle with the outside. A point on the circle itself has $OP = r$, so the relation forces $OP' = r$ and the point stays fixed. Applying the inversion twice returns every point to where it began, so inversion is its own inverse.

The map is useful because of what it does to whole figures: it carries lines and circles to lines and circles. A configuration of circles can therefore be inverted into a configuration of straight lines, studied there, and inverted back.

Ways to work on it

• Walkthrough. The defining relation, fixed points, and the line-circle interchange.
• Practice. Use the defining relation to invert a point through a circle.
• Hardest. Invert two points and derive the distance-distortion formula.

Not sure where to start? Take the ten-question placement test.