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Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL₂[real number]

Author: Vladimir V Kisil
Publisher: London : Imperial College Press ; Singapore : Distributed by World Scientific, ©2012.
Edition/Format:   Print book : CD for computer : Document   Computer File : EnglishView all editions and formats
Database:WorldCat
Summary:
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  Read more...
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Material Type: Document
Document Type: Book, Computer File
All Authors / Contributors: Vladimir V Kisil
ISBN: 9781848168589 1848168586
OCLC Number: 793214093
Notes: DVD-ROM contains illustrations, software, documentation in .pdf format, etc.
Description: xiv, 192 pages : illustrations ; 24 cm + 1 computer optical disc (4 3/4 in.)
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 SL₂[real number]. 3.1. The affine group and the real line. 3.2. One-dimensional subgroups of SL₂[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.
Other Titles: Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL₂(R)
Responsibility: Vladimir V. Kisil.

Abstract:

Deals with the Mobius transformations of the hypercomplex plane. This title provides results about geometry of circles, parabolas and hyperbolas, with the approach based on the Erlangen program of F  Read more...

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