This book is a unique exposition of rich and inspiring geometries associated with Moebius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL2(R). Starting from elementary facts in group theory, the author unveils surprising new results about the geometry of circles, parabolas and hyperbolas, using an approach based on the Erlangen programme of F Klein, who defined geometry as a study of invariants under a transitive group action.The treatment of elliptic, parabolic and hyperbolic Moebius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers which represent all non-isomorphic commutative associative two-dimensional algebras with unit. The hypercomplex numbers are in perfect correspondence with the three types of geometries concerned. Furthermore, connections with the physics of Minkowski and Galilean space-time are considered.
Erlangen Programme: Introduction; Groups and Homogeneous Spaces; Homogeneous Spaces from the Group SL(2,R); Fillmore - Springer - Cnops Construction; Metric Invariants in Upper Half-Planes; Global Geometry of Upper Half-Planes; Conformal Unit Disk; Geodesics; Unitary Rotations.