Contents -- Preface -- List of Figures -- 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 -- 2.2.1 From a Homogeneous Space to the Isotropy Subgroup -- 2.2.2 From a Subgroup to the Homogeneous Space -- 2.3 Differentiation on Lie Groups and Lie Algebras -- 2.3.1 One-parameter Subgroups and Lie Algebras -- 2.3.2 Invariant Vector Fields and Lie Algebras -- 2.3.3 Commutator in Lie Algebras -- 3. Homogeneous Spaces from the Group SL2(R) -- 3.1 The Affine Group and the Real Line -- 3.2 One-dimensional Subgroups of SL2(R) -- 3.3 Two-dimensional Homogeneous Spaces -- 3.3.1 From the Subgroup K -- 3.3.2 From the Subgroup N -- 3.3.3 From the Subgroup A -- 3.3.4 Unifying All Three Cases -- 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 -- 4.4.1 Projective Coordiantes and Polynomials -- 4.4.2 Co-Adjoint Representation -- 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 -- 8.3.1 Optics -- 8.3.2 Classical Mechanics -- 8.3.3 Quantum 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 -- 10.4.1 Cayley Transform and FSSc -- 10.4.2 Geodesics on the Disks -- 11. Unitary Rotations -- 11.1 Unitary Rotations -An Algebraic Approach -- 11.2 Unitary Rotations -A Geometrical Viewpoint -- 11.3 Rebuilding Algebraic Structures from Geometry -- 11.3.1 Modulus and Argument -- 11.3.2 Rotation as Multiplication -- 11.4 Invariant Linear Algebra -- 11.4.1 Tropical form -- 11.4.2 Exotic form -- 11.5 Linearisation of the Exotic Form -- 11.6 Conformality and Geodesics -- 11.6.1 Retrospective: Parabolic Conformality -- 11.6.2 Perspective: Parabolic Geodesics -- Epilogue: About the Cover -- Appendix A Supplementary Material -- A.1 Dual and Double Numbers -- A.2 Classical Properties of Conic Sections -- A.3 Comparison with Yaglom's Book -- A.4 Other Approaches and Results -- A.5 FSCc with Clifford Algebras -- Appendix B How to Use the Software.

B.1 Viewing Colour Graphics -- B.2 Installation of CAS -- B.2.1 Booting from the DVD Disk -- B.2.2 Running a Linux Emulator -- B.2.3 Recompiling the CAS on Your OS -- B.3 Using the CAS and Computer Exercises -- B.3.1 Warming Up -- B.3.2 Drawing Cycles -- B.3.3 Further Usage -- B.4 Library for Cycles -- B.5 Predefined Objects at Initialisation -- Bibliography -- Index.