On the Quaternionic Focal Curves

• Main Authors: Nurten (BAYRAK) GÜRSES, Özcan BEKTAŞ, Salim YÜCE
• Format: Article
• Language: English
• Published: Suleyman Demirel University 2017-06-01
• Series: Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi
• Subjects: Quaternions; Quaternionic curves; Osculating hypersphere; Focal curves; Semi-Euclidean space
• Online Access: http://dergipark.ulakbim.gov.tr/sdufenbed/article/view/5000208064

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.