Spatial Quaternionic and Quaternionic Osculating Direction Curve

Abstract

In this paper, we investigate a new structure of unit speed associated curves, such as spatial quaternionic and quaternionic osculating direction curves. For this, we assume that the vector fields 휒.%/ = 휐1.%/t.%/+ 휐2.%/n.%/+ 휐3.%/b.%/ where 휐2 1.%/+ 휐2 2.%/ = 1 for the spatial quaternionic curve and 휒.%/ = 휆1.%/æ.%/ + 휆2.%/휂.%/ + 휆3훽2.%/, where 휆2 1.%/+휆2 2.%/+휆2 3.%/ = 1 for the quaternionic curve 휙. Then, we give the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we examine in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. Finally, we give the examples and draw the figures of curves in the examples.