An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library
arXiv:1512.02960 · doi:10.15673/tmgc.v11i3.1203
Abstract
We propose to consider 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 Moebius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterise many other conformally-invariant objects, e.g. loxodromes or continued fractions. The paper describes 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. To show its usefulness, the method is implemented as a C++ library. It 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.
LaTeX 16pp+111pp of appendices, including 10 PDF graphic files and program code; v2: major revision of the paper, code in v3.1; v3: formal definition of the extended geometry, connection with integrable systems, code in v3.2rc1
References in corpus (5)
- Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group
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Cited by in corpus (5)
- Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group
- Poincare Extension of Moebius Transformations
- Cycles Cross Ratio: an Invitation
- MoebInv: C++ libraries for manipulations in non-Euclidean geometry
- Conformal Parametrisation of Loxodromes by Triples of Circles