Abstract
In this work, we firstly introduce notions of principal directed curves and principal donor curves which are associated curves of a Frenet curve in the dual Lorentzian space D 3 1 D13 . We give some relations between the curvature and the torsion of a dual principal directed curve and the curvature and the torsion of a dual principal donor curve. We show that the dual principal directed curve of a dual general helix is a plane curve and obtain the equation of dual general helix by using position vector of plane curve. Then we show that the principal donor curve of a circle in $\mathbb{D}^{2}$ or a hyperbola in $\mathbb{D}_{1}^{2}$ and the principal directed curve of a slant helix in $\mathbb{D}_{1}^{3}$ are a helix and general helix, respectively. We explain with an example for the second case. Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .
Highlights
It is very interesting to study curves in both dual space D3 and dual Lorentzian space D31
We examine associated curves of a Frenet curve in D31 and show these curves to correspond to any timelike or spacelike ruled surfaces in Minkowski
We show that the principal donor curve of a circle in D2 or a hyperbola in D21 is a dual helix and we obtain that the principal directed curve of a dual slant helix is a dual general helix
Summary
It is very interesting to study curves in both dual space D3 and dual Lorentzian space D31. Because a differentiable curve on dual unit sphere in D3 represents a ruled surface in Euclidean 3−space R3 with the aid of the E. A differentiable curve on dual pseudo hyperbolic space H20 in D31 corresponds to a timelike ruled surface in Minkowski 3−space R31 and the timelike Dual Lorentzian space, associated curves, dual general helix, dual slant helix, principal directed rectifying curve, ruled surface. The dual arc length of the dual curve γ is given by s s s s = ∥γ(σ)′∥ dσ = ∥γ(σ)′∥ dσ + ξ < t, γ∗(σ) > dσ = s + ξs∗, where s and t is arclength and the unit tangent vector of γ, respectively. According to causal character of the principal donor curve of a principal directed rectifying curve in D31, we show that this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R31
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
