The groups of two by two matrices in double and dual numbers, and associated Möbius transformations

Mustafa, Khawlah A.

doi:10.48550/arxiv.1707.01349

Cited by 2 publications

(4 citation statements)

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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%

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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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Section: 2mentioning

confidence: 99%

Section: Mathematical Usage Of Libraries Cycle and Figurementioning

confidence: 99%

Möbius--Lie Geometry and Its Extension

Kisil¹

2019

GIQ

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show abstract

“…It is common [12,14,25,26,51] 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 [34,36,48,58] and physics [20-22, 37, 38, 43] as well. Yet, the geometry with idempotent units in spaces with dimensionality n > 2 is still not sufficiently elaborated.…”

Section: Möbius-lie Geometry and The Cycle Librarymentioning

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 [30,33,34]. Some advances in the two-dimensional space were achieved recently [36,48], however further developments in higher dimensions are still awaiting their researchers.…”

Section: Mathematical Usage Of the Librarymentioning

confidence: 99%

An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library

Kisil

2015

Preprint

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We propose to consider ensembles of cycles (quadrics), which are interconnected through conformalinvariant 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 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. ContentsList of Figures 1. Introduction 2. Möbius-Lie Geometry and the cycle Library 2.1. Möbius-Lie geometry and FSC construction 2.2. Clifford algebras, FLT transformations, and Cycles 3. Ensembles of Interrelated Cycles and the figure Library 3.1. Connecting quadrics and cycles 3.2. Figures as families of cycles-functional approach 4. Mathematical Usage of the Library 5. To Do List Acknowledgement References Appendix A. Examples of Usage A.1. Hello, Cycle! A.2. Animated cycle A.3. An illustration of the modular group action A.4. Simple analysitcal demonstration A.5. The nine-points theorem-conformal version A.6. Proving the theorem: Symbolic computations A.7. Numerical relations A.8. Three-dimensional examples Appendix B. Public Methods in the figure class B.1. Creation and re-setting of figure, changing metric B.2. Adding elements to figure B.3. Modification, deletion and searches of nodes B.4. Check relations and measure parameters B.5. Accessing elements of the figure B.6. Drawing and printing B.7. Saving and openning Appendix C. Public methods in cycle relation Appendix D.

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The groups of two by two matrices in double and dual numbers, and associated Möbius transformations

Cited by 2 publications

References 0 publications

Möbius--Lie Geometry and Its Extension

Möbius--Lie Geometry and Its Extension

An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library

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