The Groups of Two by Two Matrices in Double and Dual Numbers, and Associated Möbius Transformations

Mustafa, Khawlah A.

doi:10.1007/s00006-018-0910-7

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…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]. Some advances in the two-dimensional space were achieved recently [13,37], however further developments in higher dimensions are still awaiting their researchers.…”

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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“…It is common [20,22,33,34,67] to consider mainly Clifford algebras C (n) = C (n, 0, 0) of the Euclidean space or the algebra C (p, q) = C (p, q, 0) of the pseudo-Euclidean (Minkowski) spaces. However, Clifford algebras C (p, q, r), r > 0 with nilpotent generators e 2 i = 0 correspond to interesting geometry [44,46,60,77] and physics [28-30, 47, 48, 53] as well. Yet, the geometry with idempotent units in spaces with dimensionality n > 2 is still not sufficiently elaborated.…”

Section: 2mentioning

confidence: 99%

“…Computer experiments are especially valuable for Lie geometry of indefinite or nilpotent metrics since our intuition is not elaborated there in contrast to the Euclidean space [40,43,44]. Some advances in the two-dimensional space were achieved recently [46,60], however further developments in higher dimensions are still awaiting their researchers.…”

Section: Mathematical Usage Of Libraries Cycle and Figurementioning

confidence: 99%

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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The Fixed Points and Cross-Ratios of Hyperbolic Möbius Transformations in Bicomplex Space

Chen

Dai

2022

Adv. Appl. Clifford Algebras

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The Groups of Two by Two Matrices in Double and Dual Numbers, and Associated Möbius Transformations

Cited by 5 publications

References 34 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

The Fixed Points and Cross-Ratios of Hyperbolic Möbius Transformations in Bicomplex Space

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