Circles in projective geometry

30 December 2008 at 12:27 pm 1 comment

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 \mathbb P^1(\mathbb C) is a natural compactification of the Euclidean plane: its natural automorphisms are homographies or Möbius transformations, z \mapsto \frac{az+b}{cz+d} 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 SO_2(\mathbb R^2) \ltimes \mathbb R^2.

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

SL_2(\mathbb C) / SL_2(\mathbb R) = SL(2)/SU(1,1)

which is the set of choices of an hermitian metric of signature (1,1) on \mathbb C^2. Indeed, a circle or a line can always be described by equations of the form

|a(z-z_1)| = |b(z-z_2)|

where z_1 \neq z_2 (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 PSL_2 to the affine orthogonal group SO_2 \ltimes \mathbb A^2. Since SO_2 is isomorphic to the group of matrices \begin{pmatrix} \exp(i\theta) & 0 \\ 0 & \exp(-i\theta) \\ \end{pmatrix}, 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 {[1:i:0]} and {[1:-i:0]}, corresponding to the equation x^2+y^2 = z= 0. 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 z^2 = 0.


Entry filed under: english, euclidean geometry, geometry, projective geometry. Tags: , , , .

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1 Comment Add your own

  • 1. Claude Fabre  |  2 December 2009 at 2:58 pm

    Bien que je sois un vieux monsieur (qui a fait ses études au lycée Masséna) je viens de passer mon doctorat (géométrie hyperbolique) à la Technische Unviversität de Berlin.

    Je viens de commencer à étudier le livre de Darboux sur les courbes et surfaces algébriques. N’ayant pas pratiqué la géométrie projective, je surfais donc pour me tuyauter sur les points cycliques et autres créatures étranges du CP2.

    Bravo pour ton article, concis, compétent, clair.
    Bravo aussi pour ta page.

    Si tu peux me recommander des lectures je t’en serai reconnaissant.

    Quel est le sujet de ta thèse?

    Bon courage.



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