Lie Sphere Geometry - Basic Concepts

Basic Concepts

The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing. This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius). This extension is known as Möbius geometry. Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations.

These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space of dimension 4 or 5, which is known as the Lie quadric. The natural symmetries of this quadric form a group of transformations known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric, not of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the Laguerre transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the affine conformal maps.