Circle geometry in higher dimensions. II

Publisher Summary This chapter presents a definition of a d-dimensional circle-geometry for each integer d ≥2. The definition is framed in such a manner that the circle-geometries of dimension 2 are precisely the inversive planes and that the circle-geometries of dimension d embrace all regular spreads of (d − l)-dimensional projective subspaces of a (2d − l)-dimensional projective space. Inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line. Many difficult problems in geometry become much more tractable when an inversion is applied.