Subject(s) 
Möbius transformations


Transformations (Mathematics)

Physical Description 
xiv, 192 p. : ill. ; 24 cm. + 1 computer optical disc (4 3/4 in.) 
Note 
INCLUDES DVDROM, FILED IN DISK STORAGE AT CIRCULATION DESK. 

DVDROM contains illustrations, software, documentation in .pdf format, etc 

Includes bibliographical references (p. 173179) and index 
Contents 
[v. 1.] Text  [v. 2.] DVDROM. 

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 
NOTE 
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