Posts tagged ‘Riemann sphere’

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 \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.


30 December 2008 at 12:27 pm 1 comment