MoebInv notebooks

This is a collections of Jupyter notebooks on Lie-Möbius sphere geometry [3] and MoebInv package [4]. We discuss and discover mathematical aspects of the theory using the dedicated software. The project is hosted on GitHub.

See the annotated Table of Contents as a Jupyter notebook.

Running MoebInv notebooks

The code from these notebooks can be executed at Google CoLab after the installation of MoebInv library by a couple of mouse clicks.

Specifically, open the notebook with library installation instructions and execute one cells with scripts. Then you can run all other notebooks until your CoLab session will expire.

Alternatively these notebooks can be executed from a CodeOcean capsule.

Introductory examples

• Euclidean and Lobachevsky lines This notebook also show how to make initial software installation

  ⌘ View on Github, or 👁 HTML view or ⚙ Execute on Colab.

• Nine point theorem: Some non-trivial illustrations on the celebrated result

  ⌘ View on Github, or 👁 HTML view or ⚙ Execute on CoLab.

• Example of symbolic computations Analytic proof of a simple geometric statement

  ⌘ View on Github or 👁 HTML view or ⚙ Execute on CoLab.

• What is PyGiNaC? Find out what Math can be done with it.

  ⌘ View on GitHub or 👁 HTML view or ⚙ Execute on CoLab.

Main folders

• Introduction – a collection of quick-start examples

• Geometry_of_cycles – main collection of notebooks

• EPAL-v1 – Notebooks with all computer-assisted solutions of exercises from [2].

• Appendix: illustrations – some examples of MoebInv usage which generate attractive graphics.

Jupyter and Yaglom integration

There is a Graphical User Interface (GUI) to the MoebInv library. It allows to create and research geometrical constructions by mouse clicks. You may find a pre-compiled GUI binary distribution for your desktop platform at SourceForge.

There is bidirectional integration beetween Jupyter and GUI:

• From Jupyter (or any coding system in C++ or Python) use

F.save("my-figure-archive.gar")

to save GiNaC Gar archive. Later it can be loaded into GUI (or any other front-end, indeed).

• From GUI select

File → Export figure → Export Python script

in Main Menu and save the script file. Then it can be executed in the IPython or Jupyter cell.

Furthermore, to use such code as Jupyter notebook I recommend to post-process the generated Python script with p2j (Python to Jupyter) utility.

Here is an Example or its 👁 HTML view of a Python script and Jupyter notebook automatically created from the GUI.

### References

1. Vladimir V. Kisil, Fillmore–Springer–Cnops Construction Implemented in GiNaC, Advances in Applied Clifford Algebras, 17(2007), no. 1, pp. 59–70, on-line arXiv:cs.MS/0512073.

2. Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL(2, R). Imperial College Press, London, 2012. Includes a live DVD.

3. Vladimir V. Kisil, An Extension of Mobius–Lie Geometry With Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library, Proc. Int. Geom. Cent., 11(2018), n.3, pp.45–67, on-line arXiv:1512.02960

4. Vladimir V. Kisil, MoebInv: C++ Libraries for Manipulations in Non-Euclidean Geometry, SoftwareX, 11(2020), 100385. arXiv:1912.03489.