The Resource Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2[real number], Vladimir V. Kisil

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

Label
Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2[real number]
Title
Geometry of Möbius transformations
Title remainder
elliptic, parabolic and hyperbolic actions of SL2[real number]
Statement of responsibility
Vladimir V. Kisil
Title variation
Geometry of Möbius transformations
Title variation remainder
elliptic, parabolic and hyperbolic actions of SL2(R)
Creator
Subject
Language
eng
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 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
Cataloging source
UKMGB
http://library.link/vocab/creatorName
Kisil, Vladimir V
Illustrations
illustrations
Index
index present
LC call number
QA601
LC item number
.K57 2012
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/subjectName
  • Möbius transformations
  • Transformations (Mathematics)
Label
Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2[real number], Vladimir V. Kisil
Instantiates
Publication
Note
DVD-ROM contains illustrations, software, documentation in .pdf format, etc
Accompanying material
1 DVD-ROM (4 3/4 in.)
Bibliography note
Includes bibliographical references (p. 173-179) 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. 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
Dimensions
24 cm. +
Extent
xiv, 192 p.
Isbn
9781848168589
Isbn Type
(hbk.)
Lccn
^^2012382005
Other physical details
ill.
System control number
  • (OCoLC)793214093
  • (OCoLC)ocn793214093
Label
Geometry of Möbius transformations : elliptic, parabolic and hyperbolic actions of SL2[real number], Vladimir V. Kisil
Publication
Note
DVD-ROM contains illustrations, software, documentation in .pdf format, etc
Accompanying material
1 DVD-ROM (4 3/4 in.)
Bibliography note
Includes bibliographical references (p. 173-179) 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. 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
Dimensions
24 cm. +
Extent
xiv, 192 p.
Isbn
9781848168589
Isbn Type
(hbk.)
Lccn
^^2012382005
Other physical details
ill.
System control number
  • (OCoLC)793214093
  • (OCoLC)ocn793214093

Library Locations

  • African Studies LibraryBorrow it
    771 Commonwealth Avenue, 6th Floor, Boston, MA, 02215, US
    42.350723 -71.108227
  • Alumni Medical LibraryBorrow it
    72 East Concord Street, Boston, MA, 02118, US
    42.336388 -71.072393
  • Astronomy LibraryBorrow it
    725 Commonwealth Avenue, 6th Floor, Boston, MA, 02445, US
    42.350259 -71.105717
  • Fineman and Pappas Law LibrariesBorrow it
    765 Commonwealth Avenue, Boston, MA, 02215, US
    42.350979 -71.107023
  • Frederick S. Pardee Management LibraryBorrow it
    595 Commonwealth Avenue, Boston, MA, 02215, US
    42.349626 -71.099547
  • Howard Gotlieb Archival Research CenterBorrow it
    771 Commonwealth Avenue, 5th Floor, Boston, MA, 02215, US
    42.350723 -71.108227
  • Mugar Memorial LibraryBorrow it
    771 Commonwealth Avenue, Boston, MA, 02215, US
    42.350723 -71.108227
  • Music LibraryBorrow it
    771 Commonwealth Avenue, 2nd Floor, Boston, MA, 02215, US
    42.350723 -71.108227
  • Pikering Educational Resources LibraryBorrow it
    2 Silber Way, Boston, MA, 02215, US
    42.349804 -71.101425
  • School of Theology LibraryBorrow it
    745 Commonwealth Avenue, 2nd Floor, Boston, MA, 02215, US
    42.350494 -71.107235
  • Science & Engineering LibraryBorrow it
    38 Cummington Mall, Boston, MA, 02215, US
    42.348472 -71.102257
  • Stone Science LibraryBorrow it
    675 Commonwealth Avenue, Boston, MA, 02445, US
    42.350103 -71.103784
Processing Feedback ...