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.

In this study, the planar kinematics is studied in generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the one parameter planar kinematic formulas for homothetic motions in the generalized complex plane mentioned briefly. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in generalized complex plane are obtained during the one parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, is expressed for homothetic motions in generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.