On the Quaternionic Focal Curves

Year 2017, Volume: 21 Issue: 2, 357 - 366, 13.06.2017

Nurten (bayrak) Gürses , Özcan Bektaş , Salim Yüce

Abstract

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.

Keywords

Quaternions , Quaternionic curves; Osculating hypersphere; Focal curves; Semi-Euclidean space

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