Table of Contents of
Exercises
This folder contains notebooks for computer-assisted solutions of
some exercises from [1].
It is also a comprehensive collection of tests of MoebInv::cycle library
functionality and demonstration-by-example of its usage.
- ex.I.3.7: Check formulas for Moebius action
- ex.I.3.13.i: We define the Moebius transformation
with the K subgroup
- ex.I.3.13.iv: Transverse lines for K-orbits
- ex.I.3.16: First unification of EPH geometries in
3D and rotations of a cone
- ex.I.3.21.i: Derived action of subgroups A’ and N’
- ex.I.3.25: We check that trigonometric formulae
- th.I.4.13: Proving conjugation formula for
Fillmore-Springer-Cnops construction
- ex.I.4.15: Print explicit image of a cycle under
conjugation
- ex.I.4.16: Orbits of A, N and K subgroups as cycles
and invariant transversal
- ex.I.4.20.iii: Transformations of the symplectic
form under matrix multiplication
- ex.I.5.9: Various examples of pencils spanned by
two cycles
- ex.I.5.12.i: Cycle product is invariant under the
SL(2,R) conjugation
- ex.I.5.12.ii: Explicit expression for cycle product
- ex.I.5.12.iii: Cycle product for cycles from
centres and radii
- ex.I.5.17.iv: Cycle product of two zero-radius
cycles
- ex.I.5.18.iv: We check properties of h-zero-radius
cycles
- ex.I.5.18.v: Elliptic centre of the image of a zero
radius cycle with the centre (u,v)
- ex.I.5.21: Power of point through the cycle inner
product
- ex.I.5.23.i: Check the formula for an inversive
distance for circles
- ex.I.5.23.ii: Power of intersecting cycles and
cosine of intersection angle
- ex.I.5.24: Calculation of the isotropy subgroup
orbits
- ex.I.5.25: Types of the Cauchy-Schwartz inequality
is inherited by the linear span in Krein spaces
- ex.I.5.28.i: sigma-drawing of certain
bsigma-zero-radius pencils
- ex.I.5.31: Relations between tangency and
parallelity
- ex.I.6.3: Geometric orthogonality condition for
circles and hyperbolas
- ex.I.6.5.i: Check the geometric orthogonality
condition for circles and hyperbolas
- ex.I.6.18: Verify basic properties of ghost cycle
- pr.I.6.19: Check the orthogonality through the
ghost cycle
- ex.I.6.22: Check expressions for cycle similarity
- ex.I.6.24.i: Check that cycle similarity respects
cycle product
- ex.I.6.24.ii: Check that the image of zero-radius
cycle is a zero-radius cycle
- ex.I.6.24.iv: There is no a suitable point space
map in parabolic case for cycle conjugation
- ex.I.6.25: Images of the rectangular grid under the
inversion
- ex.I.6.26: Check properties of reflection with
respect to orthogonality
- ex.I.6.28: Check that all orthogonal cycles passing
a point meet at its invesion
- ex.I.6.31: Properties of conjugation of a real line
to a certain cycle
- ex.I.6.32: A cycle passing the centre of inversion
has a straight line as its image
- ex.I.6.33: Check Yaglom inversion properties
- ex.I.6.38: Cycle similarity of the real line
- ex.I.6.39: The explicit formula for f-orthogonality
- pr.I.6.40: Proposition about f-ghost cycle
- ex.I.7.3: Check basic formulae for determinant
- le.I.7.5: Lemma: eplicit formula for the distance
between two points
- ex.I.7.7: First we define the distance of functions
- ex.I.7.8: Check the formula for an inversive
distance for a generic cycle
- le.I.7.11.i: Lemma: explicit expressions for
lengths from center
- le.I.7.11.ii: Lemma: the length from the cycle with
a focus at one point.
- ex.I.7.14.i: Conformality of distances
- ex.I.7.14.ii: Conformality of lengths
- ex.I.7.19: Calculate explicit formulas for
perpendicular
- ex.I.7.23: Check basic properties of the
infinitesimal radius cycle
- ex.I.7.24: Infinitesimal cycle is the locus of
infinitesimally close points
- ex.I.7.25: Invariance of family of infinitesimal
radius cycles under group and cycle conjugation
- ex.I.7.26: Infinitesimal radius cycle and
orthogonality condition
- ex.I.7.29: Focus of an infinitesimal cycle is
mapped by SL(2,R) to the (almost) focus of image
- ex.I.7.30: We check (the absence of) infinitesimal
conformality of parabolic lengths
- ex.I.8.2: Check properties of the zero-radius cycle
at infinity
- ex.I.9.5.iii: We check invariance of Lobachevski
metric
- ex.I.9.20: Check the basic properties of geodesics
- ex.I.10.9: We verify the expression of the
parabolic Cayley transform on the cycles
- ex.I.10.11: Images of A- and K-orbits under the
parabolic Cayley transform
- ex.I.10.12: The family of infinitesimal cycles is
invariant under the Cayley transform
- ex.I.10.13: Check relation between parabolic Cayley
transform and f-orthogonality
- ex.I.10.14: Check geodesics under the cayley
transform
- pr.I.11.24: We check conformality of distances and
lengths
This notebook is a part of the MoebInv notebooks project [2].
References
1. Vladimir V. Kisil.
Geometry of Möbius Transformations: Elliptic, Parabolic and
Hyperbolic Actions of SL2(ℝ). Imperial
College Press, London, 2012. Includes a live DVD.
2. Vladimir V. Kisil, MoebInv
notebooks, 2019.
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