Directional spherical indicatrices of timelike space curve

Abstract

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

Geometry of Solutions of the Quasi-Vortex Filament Equation in Euclidean 3-Space E3

“…By substituting from the above equation in ( 20) and ( 21), we get the functions ρ, σ, and ω given in (17). The proof is complete.…”

“…where the functions ρ, σ, ω are given in (17). Thus, from ( 13), the quasi-Gaussian curvature, quasi-mean curvature, and quasi-principal curvatures can be expressed as in (23), respectively.…”

“…where the functions ρ, σ, and ω are given in (17). Thus, from (14), we obtain the quasinormal curvature as…”

“…where the functions ρ, σ, and ω are given in (17). Thus, from (14), we obtain the quasigeodesic torsion as T g = 1 κ 2 κ 1 κ 2s − κ 1s κ 2 + κ 3 κ 2 .…”

“…At all locations, the Q-frame is well-defined, it is straightforward to calculate, and its construction is unaffected by whether the curve is parameterized by arc-length or not. Among many papers dealing with this subject, we refer the reader to the following [16][17][18].…”