Some relations between rectifying and normal curves in Minkowski 3-space

In this paper, we obtain explicit parameter equations of spacelike rectifying curves in $E^3_1$ whose projection onto spacelike, timelike and lightlike plane of $E^3_1$ is a normal curve. We also obtain explicit parameter equations of spacelike normal curves in $E^3_1$ whose projection onto lightlike plane of $E^3_1$, with respect to a chosen screen distribution, is a rectifying W-curve

On eqiform Darboux helices in Galilean 3-space

In this paper, we define equiform Darboux helices in a Galilean space (G_{3}) and obtain their explicit parameter equations. We show that equiform Darboux helices have only a non-isotropic axis and characterize equiform Darboux vectors of equiform Darboux helices in terms of equiform rectifying curves. We prove that an equiform Darboux vector of an equiform Darboux helix α is an equiform Darboux helix if an admissible curve (alpha) is a rectifying curve. We also prove that there are no equiform curves of constant precession and give some examples of equiform Darboux helices

k-type null slant helices in Minkowski space-time

In this paper, we introduce a notion of a $k$-type null slant helices inMinkowski space-time, where $kin {0,1,2,3}$. We give the necessary andsufficient conditions for null Cartan curves to be $k$-type null slanthelices in terms of their curvatures $kappa _{2}$ and $kappa _{3}$. Inparticular, we characterize $k$-type null slant helices lying inpseudosphere $S^{3}_{1}(r)$. We find the relationships between $0$-type and $$-type null slant helices, as well as between $1$-type and $2$-type nullslant helices. Moreover, we prove that there are no $3$-type null slanthelices in Minkowski space-time

On involutes of order k of a null cartan curve in minkowski spaces

© 2019, University of Nis. All rights reserved. In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space En for n ≥ 3 and 1 ≤ k ≤ n − 1. In relation to that, we prove that if a null Cartan helix has 1 a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ≤ k ≤ 2. We show that among all null Cartan curves in E3, only the null Cartan cubic has two families of involutes of order 1, one of which 1 lies on B-scroll. We also give some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space. As an application, we show that involutes of order 1 of a null Cartan curve in E3 1 evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces

On k-type spacelike slant helices lying on lightlike surfaces

© University of Nis. All rights reserved. In this paper, we define k-type spacelike slant helices lying on a lightlike surface in Minkowski space E3 according to their Darboux frame for k ∈ {0, 1, 2}. We obtain the necessary and the sufficient 1 conditions for spacelike curves with non-null and null principal normal lying on lightlike surface to be the k-type spacelike slant helices in terms of their geodesic curvature, normal curvature and geodesic torsion. Additionally, we determine their axes and show that the Darboux frame of a spacelike curve lying on a lightlike surface coincides with its Bishop frame if and only if it has zero geodesic torsion. Finally, we give some examples

On t-slant, n-slant and b-slant helices in pseudo-galilean space G¹<inf>3</inf>

© 2018, University of Nis. All rights reserved. In this paper, we introduce T-slant, N-slant and B-slant helices in the pseudo-Galilean space G13 and define an angle between the spacelike and the timelike isotropic vector lying in the pseudo-Euclidean plane x = 0. In particular, we obtain the explicit parameter equations of the T-slant helices and prove that there are no N-slant and B-slant helices in G13. We also prove that there are no Darboux helices in the same space

A note on Lamarle formula in Minkowski 3-space

© 2018 Tamkang University. All rights reserved. The Lamarle formula is known as a simple relation between the Gaussian curvature and the distribution parameter of a non-developable ruled surface. In this paper, we obtain the Lamarle formula of a non-developable ruled surface with pseudo null base curve and null director vector field in Minkowski 3-space. We also obtain the corresponding striction line and distribution parameter of such surface. We prove that there is no Lamarle formula when the director vector field is spacelike and its derivative is null, because the ruled surface in that case is a lightlike plane. Finally, we give some examples