Evolution of quaternionic curve in the semi‐Euclidean space E24

Abstract

In this paper, kinematics of semi‐real quaternionic curve in semi‐Euclidean space double-struckE24 is obtained in terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic curve are obtained. Also, examples of evolution equations of curvatures are given.

A primer on the differential geometry of quaternionic curves

“…In terms of the Cartesian notation, ⟨q, e i q⟩ = 0 where e i = {i, 𝑗, k}, (12) and thus, we have that quaternions can be described as a four-dimensional real vector space. In the polar notations ( 5) and ( 6), we respectively obtain ⟨q, 𝜔q⟩ = 0, and ⟨q, q𝑗⟩ = 0, (13) and thus, the quaternions are described as two-dimensional real vector spaces. Considering that the space had originally four dimensions from (12), this conclusion seems to contradict our expectations.…”

“…The study of quaternionic curves begins in Bharathi and Nagaraj 1 where Frenet–Serret equations were determined. Several contributions have been made since then, and even the Frenet–Serret equations were further studied, 2,3 although different quaternic curves where this approach has been applied were considered as well 4–14 . These articles have the common feature that the Frenet–Serret equations are strongly influenced by the geometric description, where a local basis is determined in each point of an arbitrary smooth and regular curve.…”

On the Framed Normal Curves in Euclidean 4-space

“…In particular, timelike normal curves lie in the pseudosphere in R 3 1 . In addition, the characterizations of timelike and spacelike normal curves in Minkowski space R 3 1 have been examined recently, [12], [13]. Bahar et.…”

“…al obtained a moving adapted frame to investigate the properties of rectifying curve with singular points in R 3 , and this frame was used to analyze some special curves with singular points [2]. For more details on the notion of framed curves, see [3]- [9].…”

“…Therefore, the position vector of such curves with respect to some chosen origin always lies in its normal plane [11]. In particular, timelike normal curves lie in the pseudosphere in R 3 1 . In addition, the characterizations of timelike and spacelike normal curves in Minkowski space R 3 1 have been examined recently, [12], [13].…”