On quaternionic product on some sets of hyperbolas

The aim of this paper is to introduce some products, induced by the quaternionic product, on the set of equilateral hyperbolas. The study is a mix of elements from Euclidean and projective geometry. Some properties of these products are highlighted and are connected with some special numbers as root...

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Published inITM Web of Conferences Vol. 34; p. 3004
Main Authors Crasmareanu, Mircea, Popescu, Marcela
Format Journal Article Conference Proceeding
LanguageEnglish
Published Les Ulis EDP Sciences 2020
Subjects
Conics
equilateral hyperbola
Euclidean geometry
Hyperbolas
Isomorphism
octonion
product
Projective geometry
quaternion
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Abstract The aim of this paper is to introduce some products, induced by the quaternionic product, on the set of equilateral hyperbolas. The study is a mix of elements from Euclidean and projective geometry. Some properties of these products are highlighted and are connected with some special numbers as roots or powers of the unit. Then we extend these products in a natural manner to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas, using the algebra of octonions. Further, using an inversion, we extend these products to Bernoulli lemniscates and to Bernoulli q-lemniscates. Finally, using some isomorphisms, we extend these products to conics, and using an inversion, we extend these products to other curves. In order to highlight the importance of these studied products, the paper ends with some applications.
AbstractList The aim of this paper is to introduce some products, induced by the quaternionic product, on the set of equilateral hyperbolas. The study is a mix of elements from Euclidean and projective geometry. Some properties of these products are highlighted and are connected with some special numbers as roots or powers of the unit. Then we extend these products in a natural manner to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas, using the algebra of octonions. Further, using an inversion, we extend these products to Bernoulli lemniscates and to Bernoulli q-lemniscates. Finally, using some isomorphisms, we extend these products to conics, and using an inversion, we extend these products to other curves. In order to highlight the importance of these studied products, the paper ends with some applications.
Author Popescu, Marcela
Crasmareanu, Mircea
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Cites_doi 10.1080/0025570X.2004.11953223
10.1007/s00283-014-9445-5
10.1090/spec/054/07
ContentType Journal Article
Conference Proceeding
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References R3
Akopyan (R1) 2014; 36
Hre¸tcanu (R6) 2013; 54
R5
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Özdemyr (R7) 2011; 35
Crasmareanu (R4) 2020; 66
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SubjectTerms Conics
equilateral hyperbola
Euclidean geometry
Hyperbolas
Isomorphism
octonion
product
Projective geometry
quaternion
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