SPECIAL CURVES ACCORDING TO TYPE-2 QUATERNIONIC FRAME IN 4

Abstract

Quaternions, which were defined by William Rowan Hamilton in 1843, are a number system in four-dimensional space and are analogous to complex numbers. However, quaternion multiplication is not commutative, distinguishing them from complex numbers. Quaternions are special mathematical tools used in computer science, robotics, and many other mathematical sciences. From this point of view, they also get attention in differential geometry. In particular, their characterizations given by the Serret-Frenet apparatus are challenging. For this reason, Bharathi and Nagaraj obtained Serret-Frenet formulas for spatial quaternionic and quaternionic curves in 3and 4, respectively. Inspired by this work, the geometers obtained quaternionic frames in 4-dimensional Semi-Euclidean space , dual quaternionic space, and lately type-2 quaternionic frame in 4. Their features are also examined. On the other hand, studying the characterizations of the curves is a significant research area for differential geometers because of having used various branches of applied sciences. Particularly, the features of the specially defined curves, such as Bertrand, Smarandache, rectifying, and osculating curves, are curiously studied. In four-dimensional spaces, rectifying curves are defined as a curve whose position vector fully lies in {T,N2,N3}. Similarly, the osculating curve defined as first and second kind became of having two different binormals {T,N1,N2} and {T,N1,N3}, respectively. In this study, inspired by the definitions above, we define rectifying, osculating first kind, and osculating second kind curves according to the type-2 quaternionic frame in 4. Their characterizations are also examined.