Lie sphere geometry

Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius. Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius. The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres). To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius. Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous 'line-sphere correspondence' between the space of lines and the space of spheres in 3-dimensional space. 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 inversive geometry with automorphisms known as 'Mobius transformations'. 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. These groups also have a direct physical interpretation: As pointed out by Harry Bateman, the Lie sphere transformations are identical with the spherical wave transformations that leave the form of Maxwell's equations invariant. In addition, Élie Cartan, Henri Poincaré and Wilhelm Blaschke pointed out that the Laguerre group is simply isomorphic to the Lorentz group of special relativity (see Laguerre group isomorphic to Lorentz group). Eventually, there is also an isomorphism between the Möbius group and the Lorentz group (see Möbius group#Lorentz transformation). The Lie quadric of the plane is defined as follows. Let R3,2 denote the space R5 of 5-tuples of real numbers, equipped with the signature (3,2) symmetric bilinear form defined by The projective space RP4 is the space of lines through the origin in R5 and is the space of nonzero vectors x in R5 up to scale, where x= (x0,x1,x2,x3,x4). The planar Lie quadric Q consists of the points in projective space represented by vectors x with x · x = 0.