This book is a unique exposition of rich and inspiring geometries associated with Mobius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL[symbol](real number). 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 Mobius 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.

Description-Table Of Contents

1. Erlangen programme: preview. 1.1. Make a guess in three attempts. 1.2. Covariance of FSCc. 1.3. Invariants: algebraic and geometric. 1.4. Joint invariants: orthogonality. 1.5. Higher-order joint invariants: focal orthogonality. 1.6. Distance, length and perpendicularity. 1.7. The Erlangen programme at large -- 2. Groups and homogeneous spaces. 2.1. Groups and transformations. 2.2. Subgroups and homogeneous spaces. 2.3. Differentiation on Lie groups and Lie algebras -- 3. Homogeneous spaces from the group SL[real number]. 3.1. The affine group and the real line. 3.2. One-dimensional subgroups of SL[real number]. 3.3. Two-dimensional homogeneous spaces. 3.4. Elliptic, parabolic and hyperbolic cases. 3.5. Orbits of the subgroup actions. 3.6. Unifying EPH cases: the first attempt. 3.7. Isotropy subgroups -- 4. The extended Fillmore-Springer-Cnops construction. 4.1. Invariance of cycles. 4.2. Projective spaces of cycles. 4.3. Covariance of FSCc. 4.4. Origins of FSCc. 4.5. Projective cross-ratio -- 5. Indefinite product space of cycles. 5.1. Cycles: an appearance and the essence. 5.2. Cycles as vectors. 5.3. Invariant cycle product. 5.4. Zero-radius cycles. 5.5. Cauchy-Schwarz inequality and tangent cycles -- 6. Joint invariants of cycles: orthogonality. 6.1. Orthogonality of cycles. 6.2. Orthogonality miscellanea. 6.3. Ghost cycles and orthogonality. 6.4. Actions of FSCc matrices. 6.5. Inversions and reflections in cycles. 6.6. Higher-order joint invariants: focal orthogonality -- 7. Metric invariants in upper half-planes. 7.1. Distances. 7.2. Lengths. 7.3. Conformal properties of Mobius maps. 7.4. Perpendicularity and orthogonality. 7.5. Infinitesimal-radius cycles. 7.6. Infinitesimal conformality -- 8. Global geometry of upper half-planes. 8.1. Compactification of the point space. 8.2. (Non)-invariance of the upper half-plane. 8.3. Optics and mechanics. 8.4. Relativity of space-time -- 9. Invariant metric and geodesics. 9.1. Metrics, curves' lengths and extrema. 9.2. Invariant metric. 9.3. Geodesics: additivity of metric. 9.4. Geometric invariants. 9.5. Invariant metric and cross-ratio -- 10. Conformal unit disk. 10.1. Elliptic Cayley transforms. 10.2. Hyperbolic Cayley transform. 10.3. Parabolic Cayley transforms. 10.4. Cayley transforms of cycles -- 11. Unitary rotations. 11.1. Unitary rotations - an algebraic approach. 11.2. Unitary rotations - a geometrical viewpoint. 11.3. Rebuilding algebraic structures from geometry. 11.4. Invariant linear algebra. 11.5. Linearisation of the exotic form. 11.6. Conformality and geodesics.