Laguerre Arc Length from Distance Functions

Barrett, David; Bolt, Michael

doi:10.4310/ajm.2010.v14.n2.a3

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…In the parabolic space the nine-point Theorem 4.1 is not preserved in this manner. It is already observed [41], [35], [46], [39], [43], [40], [52], [3], that the degeneracy of parabolic metric in the point space requires certain revision of traditional definitions. The parabolic variation of nine-point theorem may prompt some further considerations as well.…”

Section: Mathematical Usage Of the Librarymentioning

confidence: 99%

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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Section: Mathematical Usage Of the Librarymentioning

confidence: 99%

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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“…In most cases it corresponds to orthogonality of quadrics in the point space. More generally, most of FLT-invariant relations between quadrics may be expressed in terms FLT-invariant inner product (5). For the full description of methods on individual cycles, which are implemented in the library cycle, see the respective documentation [41].…”

Section: 2mentioning

confidence: 99%

“…By Ahlfors [2] (see also [8,§ 5;20,Thm Then MM = δI andM = κM * , where κ = 1 or −1 depending either d is a product of even or odd number of vectors.…”

Section: 5mentioning

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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Conformal Parametrisation of Loxodromes by Triples of Circles

Kisil

Reid

2019

Trends in Mathematics

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We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical examples illustrate the usage of parametrisation. Our work extends the set of objects in Lie sphere geometry-circle, lines and points-to the natural maximal conformally-invariant family, which also includes loxodromes.

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Laguerre Arc Length from Distance Functions

Cited by 4 publications

References 10 publications

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

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

Möbius--Lie Geometry and Its Extension

Conformal Parametrisation of Loxodromes by Triples of Circles

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