Circles in projective geometry
Euclidean geometry is closely related to the ability to define circles, which exists naturally in two settings: the complex projective line and the real projective plane.
The Riemann sphere
The complex projective line is a natural compactification of the Euclidean plane: its natural automorphisms are homographies or Möbius transformations, which transform lines or circles into lines or circles. The Euclidean geometry arises from the natural isomorphism between the group of homographies stabilizing the point at infinity and the group of direct isometries of the plane .
It is well-known that inscribed angles in a given circle are equal: this means the cross-ratio of the complex coordinates of 4 points on a circle is real. Circles on the Riemann sphere are thus transforms of the real projective line. More abstractly, a circle would be an element of
which is the set of choices of an hermitian metric of signature (1,1) on . Indeed, a circle or a line can always be described by equations of the form
where (equations of Apollonian circles). With this definition, circles degenerate to lines (circles with a point at infinity) or points.
The projective plane
The projective plane gets an Euclidean structure by reducing its automorphism group to the affine orthogonal group . Since is isomorphic to the group of matrices , one sees that a structure of Euclidean affine plane amounts to the choice of a line (line at infinity) and two distinct points (on the line at infinity), and to calling rotations those transformations which keeps them invariant.
This point of view was the preferred one in the 19th century for so-called analytical geometry, since it allows field extension to the complex numbers, and the study of algebraic curves other than circles.
The two distinguished (complex) points of the “Euclidean projective plane” are called cyclic points or circular points, and their traditional homogeneous coordinates are and , corresponding to the equation . They correspond to the common directions of asymptotes of circles, which can also be defined as conics through the circular points.
Thus, a circle of radius zero should be viewed as a singular conic (the union of two isotropic lines going to the circular points), and the most singular degeneration of a circle is the double line at infinity .