Generalised circle

In geometry, a generalised circle, sometimes called a cline or circline, is a straight line or a circle, the curves of constant curvature in the Euclidean plane.

The natural setting for generalised circles is the extended plane, a plane along with one point at infinity through which every straight line is considered to pass. Given any three distinct points in the extended plane, there exists precisely one generalised circle passing through all three.

Generalised circles sometimes appear in Euclidean geometry, which has a well-defined notion of distance between points, and where every circle has a center and radius: the point at infinity can be considered infinitely distant from any other point, and a line can be considered as a degenerate circle without a well-defined center and with infinite radius (zero curvature). A reflection across a line is a Euclidean isometry (distance-preserving transformation) which maps lines to lines and circles to circles; but an inversion in a circle is not, distorting distances and mapping any line to a circle passing through the reference circles's center, and vice-versa.

However, generalised circles are fundamental to inversive geometry, in which circles and lines are considered indistinguishable, the point at infinity is not distinguished from any other point, and the notions of curvature and distance between points are ignored. In inversive geometry, reflections, inversions, and more generally their compositions, called Möbius transformations, map generalised circles to generalised circles, and preserve the inversive relationships between objects.

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 generalised circles become circles on the sphere.