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Geometry of mobius transformations : elliptic, parabolic and hyperbolic actions of...
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Geometry of mobius transformations : elliptic, parabolic and hyperbolic actions of SL[symbol]([real number])
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Description
Rating
Title
Geometry
of
mobius
transformations
:
elliptic
,
parabolic
and
hyperbolic
actions
of
SL[symbol]([real
number])
Creator
Kisil, Vladimir V.
Contributors
World Scientific (Firm)
DescriptionAbstract
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
.
DescriptionTable Of 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
SL[real
number]
.
3.1
. The
affine
group
and the
real
line
.
3.2
.
Onedimensional
subgroups
of
SL[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
.
Publisher
Imperial College Press
Distributed by World Scientific Pub. Co.
Subject
Mobius transformations.
Identifier (Full text)
9781848168596
(electronic
bk.)
;
1848168586
;
9781848168589
;
http://www.worldscientific.com/worldscibooks/10.1142/P835#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xiv, 192 p. : ill.
Date
c2012
.
OCLC number
874497303
CONTENTdm number
238
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