Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2[subscript 2](R)[real number]
 Author/Creator
 Kisil, Vladimir V.
 Language
 English.
 Imprint
 London : Imperial College Press ; Singapore : Distributed by World Scientific, c2012.
 Physical description
 xiv, 192 p. : ill. ; 24 cm. + 1 DVDROM (4 3/4 in.)
Access
Available online
 www.worldscientific.com World Scientific
 ebrary

Stacks
 Library has: 1 v. + 1 DVD
ROM 
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QA601 .K57 2012
 Library has: 1 v. + 1 DVD
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Contents/Summary
 Bibliography
 Includes bibliographical references (p. 173179) and index.
 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. Higherorder 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 SL2[real number]. 3.1. The affine group and the real line. 3.2. Onedimensional subgroups of SL2[real number]. 3.3. Twodimensional 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 FillmoreSpringerCnops 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 crossratio
 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. Zeroradius cycles. 5.5. CauchySchwarz 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. Higherorder joint invariants: focal orthogonality
 7. Metric invariants in upper halfplanes. 7.1. Distances. 7.2. Lengths. 7.3. Conformal properties of Mobius maps. 7.4. Perpendicularity and orthogonality. 7.5. Infinitesimalradius cycles. 7.6. Infinitesimal conformality
 8. Global geometry of upper halfplanes. 8.1. Compactification of the point space. 8.2. (Non)invariance of the upper halfplane. 8.3. Optics and mechanics. 8.4. Relativity of spacetime
 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 crossratio
 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.
 Summary
 This book is a unique exposition of rich and inspiring geometries associated with Mobius transformations of the hypercomplex plane. The presentation is selfcontained 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 nonisomorphic commutative associative twodimensional 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 spacetime are considered.
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Vladimir V. Kisil.
 Title Variation
 Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2(R)
 Note
 DVDROM contains illustrations, software, documentation in .pdf format, etc.
 ISBN
 9781848168589
 1848168586