Möbius-Lie geometry and its extension. (English) Zbl 1415.51003

Kisil, Vladimir V.

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 20th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 2–7, 2018. Sofia: Avangard Prima; Sofia: Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering. Geom. Integrability Quantization 20, 13-61 (2019).

Summary: This paper is a review of the classical Möbius-Lie geometry and recent works on its extension. The latter considers ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. “to be orthogonal”, “to be tangent”, etc.), as new objects in an extended Möbius-Lie geometry. It is shown on examples, that such ensembles of cycles naturally parameterise many other conformally-invariant families of objects, two examples – the Poincaré extension and continued fractions are considered in detail. Further examples, e.g. loxodromes, wave fronts and integrable systems, are discussed elsewhere.
The extended Möbius-Lie geometry is efficient due to a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. The algorithmic nature of the method allows to implement it as a C\(^{++}\) library, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well.
For the entire collection see [Zbl 1408.53003].

MSC:

51B25	Lie geometries in nonlinear incidence geometry
30B70	Continued fractions; complex-analytic aspects
51M05	Euclidean geometries (general) and generalizations
51N25	Analytic geometry with other transformation groups
51B10	Möbius geometries
68U05	Computer graphics; computational geometry (digital and algorithmic aspects)
11E88	Quadratic spaces; Clifford algebras
68W30	Symbolic computation and algebraic computation

Keywords:

Clifford algebra; continued fraction; fraction-linear transformation; indefinite inner product space; integrable system; loxodrome; Möbius-Lie geometry; Poincaré extension spheres geometry

Software:

GiNaC; MoebInv; Python