Geometries and Transformations

Following Cayley and Klein, Geometries and Transformations builds on projective and inversive geometry to construct various 'linear' and 'circular' geometries. Among these are the classical real metric spaces - Euclidean, hyperbolic, elliptic, and spherical - and their unitary counterparts. In accordance with Klein's Erlanger Programm, each geometry is characterized by the groups of transformations that preserve its essential properties, such as collinearity, perpendicularity, parallelism, and congruence. Relative to a suitably chosen coordinate system, points are represented by vectors and geometric operations by matrices. Transformation groups are then exhibited as particular subgroups or quotient groups of the general linear group of invertible matrices. Discrete groups generated by reflections (Coxeter groups) and their subgroups are manifested in the symmetry of geometric figures and patterns, both finite and infinite. Certain groups of hyperbolic isometries can be identified with modular groups of linear fractional transformations over rings of real, complex, or quaternionic integers.