Conformal Parametrisation of Loxodromes by Triples of Circles

Kisil, Vladimir V.; Reid, James Andrew

doi:10.1007/978-3-030-23854-4_15

Cited by 3 publications

(19 citation statements)

References 19 publications

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“…The package accumulates more than 15 years of work and is still under active development. The design of the library figure shaped the general theoretical approach to the extension of Möbius-Lie geometry [2,3], which leaded to specific realisations in [33,34,35]. Furthermore, it shall be helpful for computer experiments in Lie sphere geometry of indefinite or nilpotent metrics since our intuition is not elaborated there in contrast to the Euclidean space [22,26,36].…”

Section: Discussionmentioning

confidence: 99%

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Kisil¹

2020

SoftwareX

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The introduced package MoebInv contains two C ++ libraries for symbolic, numeric and graphical manipulations in non-Euclidean geometry. The first library cycle implements basic geometric operations on cycles, which are the zero sets of certain polynomials of degree two. The second library figure operates on ensembles of cycles interconnected by Moebius-invariant relations: orthogonality, tangency, etc. Both libraries work in spaces with any dimension and arbitrary signatures of their metrics. Their essential functionality is accessible in interactive modes from Python/Jupyter shells and a dedicated Graphical User Interface. The latter does not require any coding skills and can be used in education. The package is tested on (and supplied for) various Linux distributions, Windows 10, Mac OS X and several cloud services.

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Section: Discussionmentioning

confidence: 99%

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Kisil¹

2020

SoftwareX

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“…FSCc is useful in consideration of the Poincaré extension of Möbius maps [52], loxodromes [54] and continued fractions [51]. In theoretical physics FSCc nicely describes conformal compactifications of various space-time models [32; 42; 46 [14,Ch.…”

Section: 1mentioning

confidence: 99%

“…It was shown recently that ensembles of cycles with certain FLT-invariant relations provide helpful parametrisations of new objects, e.g. points of the Poincaré extended space [52], loxodromes [54] or continued fractions [13,51], see Example 5.1 below for further details. Thus, we propose to extend Möbius-Lie geometry and consider ensembles of cycles as its new objects, cf.…”

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confidence: 99%

“…This paper provides conceptual foundations of such extension and demonstrates its practical implementation as a C++ library figure 1 . Interestingly, the development of this library shaped the general approach, which leads to specific realisations in [51,52,54].…”

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confidence: 99%

“…In Sec. 5.1 we present several examples of ensembles, which were already used in mathematical theories [51,52,54], then we describe how ensembles are encoded in the present library figure through the functional programming framework.…”

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Möbius--Lie Geometry and Its Extension

Kisil¹

2019

GIQ

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These lectures review the classical Möbius-Lie geometry and recent work 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 published 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.

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An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

Kisil

2019

ПМГЦ

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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 M\"obius--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 in\-vari\-ant 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 {\CPP} 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.

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Conformal Parametrisation of Loxodromes by Triples of Circles

Cited by 3 publications

References 19 publications

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Möbius--Lie Geometry and Its Extension

An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

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