On the Quaternionic Focal Curves

Authors : Nurten BAYRAK GÜRSES, Özcan BEKTAŞ, Salim YÜCE

Pages : 357-366

Doi:10.19113/sdufbed.14005

View : 61 | Download : 12

Publication Date : 2017-06-13

Article Type : Research Paper

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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