An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library
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...
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| Format: | Article |
| Language: | English |
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Odesa National University of Technology
2018-09-01
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| Series: | Pracì Mìžnarodnogo Geometričnogo Centru |
| Subjects: | |
| Online Access: | http:////journals.onaft.edu.ua/index.php/geometry/article/view/1203 |
| _version_ | 1811227327254757376 |
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| author | Vladimir Kisil |
| author_facet | Vladimir Kisil |
| author_sort | Vladimir Kisil |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-12T09:40:51Z |
| format | Article |
| id | doaj.art-39e0d5a6436b4bc5955e957484e0a14f |
| institution | Directory Open Access Journal |
| issn | 2072-9812 2409-8906 |
| language | English |
| last_indexed | 2024-04-12T09:40:51Z |
| publishDate | 2018-09-01 |
| publisher | Odesa National University of Technology |
| record_format | Article |
| series | Pracì Mìžnarodnogo Geometričnogo Centru |
| spelling | doaj.art-39e0d5a6436b4bc5955e957484e0a14f2022-12-22T03:38:04ZengOdesa National University of TechnologyPracì Mìžnarodnogo Geometričnogo Centru2072-98122409-89062018-09-0111310.15673/tmgc.v11i3.12031203An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC libraryVladimir Kisil0University of LeedsWe 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.//journals.onaft.edu.ua/index.php/geometry/article/view/1203Lie--Mobius geometry |
| spellingShingle | Vladimir Kisil An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library Pracì Mìžnarodnogo Geometričnogo Centru Lie--Mobius geometry |
| title | An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library |
| title_full | An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library |
| title_fullStr | An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library |
| title_full_unstemmed | An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library |
| title_short | An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library |
| title_sort | extension of mobius lie geometry with conformal ensembles of cycles and its implementation in a ginac library |
| topic | Lie--Mobius geometry |
| url | http:////journals.onaft.edu.ua/index.php/geometry/article/view/1203 |
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