EPAL-v1

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 $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.

                    2. Vladimir V. Kisil, MoebInv notebooks, 2019.

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