On the Quaternionic Focal Curves
In this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere. By using this definition and the quaternionic osculating hypersphere...
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Suleyman Demirel University
2017-06-01
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Online Access: | http://dergipark.ulakbim.gov.tr/sdufenbed/article/view/5000208064 |
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doaj-7fd4aa538eed4f0aa77c83da8ace77542020-11-24T22:17:22ZengSuleyman Demirel UniversitySüleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi1308-65292017-06-0121235736610.19113/sdufbed.140055000174498On the Quaternionic Focal CurvesNurten (BAYRAK) GÜRSES0Özcan BEKTAŞ1Salim YÜCE2Yıldız Teknik ÜniversitesiRecep Tayyip Erdoğan Üniversitesi Fen edebiyat Fakültesi Matematik BölümüYıldız Teknik Üniversitesi Fen Edebiyat Fakültesi, Matematik BölümüIn this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere. By using this definition and the quaternionic osculating hyperspheres of these curves, the quaternionic focal curves in the spaces $\Q$ and $\Q_\nu$ with index $\nu=\{1,2\}$ are discussed. Some relations about spatial semi-real quaternionic curves and semi-real quaternionic curves are examined by using focal curvatures and "scalar Frenet equations" between the focal curvatures. Then, the notions: such as vertex, flattenings, a symmetry point are defined for these curves. Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented.http://dergipark.ulakbim.gov.tr/sdufenbed/article/view/5000208064QuaternionsQuaternionic curvesOsculating hypersphereFocal curvesSemi-Euclidean space |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Nurten (BAYRAK) GÜRSES Özcan BEKTAŞ Salim YÜCE |
spellingShingle |
Nurten (BAYRAK) GÜRSES Özcan BEKTAŞ Salim YÜCE On the Quaternionic Focal Curves Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi Quaternions Quaternionic curves Osculating hypersphere Focal curves Semi-Euclidean space |
author_facet |
Nurten (BAYRAK) GÜRSES Özcan BEKTAŞ Salim YÜCE |
author_sort |
Nurten (BAYRAK) GÜRSES |
title |
On the Quaternionic Focal Curves |
title_short |
On the Quaternionic Focal Curves |
title_full |
On the Quaternionic Focal Curves |
title_fullStr |
On the Quaternionic Focal Curves |
title_full_unstemmed |
On the Quaternionic Focal Curves |
title_sort |
on the quaternionic focal curves |
publisher |
Suleyman Demirel University |
series |
Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi |
issn |
1308-6529 |
publishDate |
2017-06-01 |
description |
In this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere. By using this definition and the quaternionic osculating hyperspheres of these curves, the quaternionic focal curves in the spaces $\Q$ and $\Q_\nu$ with index $\nu=\{1,2\}$ are discussed. Some relations about spatial semi-real quaternionic curves and semi-real quaternionic curves are examined by using focal curvatures and "scalar Frenet equations" between the focal curvatures. Then, the notions: such as vertex, flattenings, a symmetry point are defined for these curves. Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented. |
topic |
Quaternions Quaternionic curves Osculating hypersphere Focal curves Semi-Euclidean space |
url |
http://dergipark.ulakbim.gov.tr/sdufenbed/article/view/5000208064 |
work_keys_str_mv |
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