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An extension of Möbius-Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library. (English) Zbl 1428.51001

Summary: 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 Möbius-Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of 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.

MSC:

51B25 Lie geometries in nonlinear incidence geometry
51N25 Analytic geometry with other transformation groups
51B10 Möbius geometries
51-04 Software, source code, etc. for problems pertaining to geometry
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