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This book is a unique exposition of rich and inspiring geometries associated with M bius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL(2, R). Starting from elementary facts in group theory, the author unveiled surprising new results about geometry of circles, parabolas and hyperbolas, with the approach based on the Erlangen program of F Klein who defined geometry as a study of invariants under a transitive group action. The treatment of elliptic, parabolic and hyperbolic M bius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers. They form three possible commutative associative two-dimensional algebras, which are in perfect correspondences with the three types of geometries concerned. Furthermore, connections with the physics of Minkowski and Galilean space-time are considered.