This group of conformal transformations by reciprocal radii preserves angles and transforms spheres into spheres or hyperspheres (see Möbius transformation, conformal symmetry, special conformal transformation). It is a 6-parameter group in the plane R which corresponds to the Möbius group of the extended complex plane, a 10-parameter group in space R, and a 15-parameter group in R. In R it represents only a small subset of all conformal transformations therein, whereas in R it is identical to the group of all conformal transformations (corresponding to the Möbius transformations in higher dimensions) therein, in accordance with Liouville's theorem. Conformal transformations in R were often applied to what Darboux (1873) called "pentaspherical coordinates" by relating the points to homogeneous coordinates based on five spheres.