Central Projection of Quadrics and Möbius Geometry

In this section we study the general construction of central projection of a quadric from a point onto its polar hyperplane, see, e.g., [Kle1928, Bla1954, Gie1982]. This leads to a double cover of a Cayley-Klein space in the hyperplane such that the spheres in that Cayley-Klein space correspond to hyperplanar sections of the quadric. Vice versa, a Cayley-Klein space can be lifted to a quadric in a projective space of one dimension higher, such that Cayley-Klein spheres lift to hyperplanar sections of the quadric. In this way, hyperbolic and elliptic geometry can be lifted to Möbius geometry, and Möbius geometry may be seen as the geometry of points and spheres of the hyperbolic or elliptic space, respectively. We demonstrate how the group of Möbius transformations can be decomposed into the respective isometries and scalings along concentric spheres.