Generalised Circle

Generalised Circle

A generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together.

Inversive plane geometry is formulated on the plane extended by one point at infinity. A straight line is then thought of as a circle that passes through the point at infinity. The fundamental transformations in inversive geometry, the inversions, have the property that they map generalized circles to generalized circles. Möbius transformations, which are compositions of inversions, inherit that property. These transformations do not necessarily map lines to lines and circles to circles: they can mix the two.

Inversions come in two kinds: inversions at circles and reflections at lines. Since the two have very similar properties, we combine them and talk about inversions at generalized circles.

Given any three distinct points in the extended plane, there exists precisely one generalized circle that passes through the three points.

The extended plane can be identified with the sphere using a stereographic projection. The point at infinity then becomes an ordinary point on the sphere, and all generalized circles become circles on the sphere.