In this article, we present the equiform parameter and define the equiform-Bishop frame in Minkowski 3-space E 3 1 . Additionally, we investigate the equiform-Bishop formulas of the equiform spacelike case in Minkowski 3-space.Furthermore, some results of equiform spacelike normal curves according to the equiform-Bishop frame in E 3 1 are considered. KEYWORDS equiform-Bishop frame, equiform curvatures, Minkowski Space, normal curves
PRELIMINARIESThe Minkowski space E 3 1 is the space R 3 , equipped with the metric g, where g is given bywhere (x 1 , x 2 , x 3 ) is a coordinate system of E 3 1 . Let v be any vector in E 3 1 , then the vector v is spacelike, timelike, or null (lightlike) if g(v, v) > 0 or v = 0, g(v, v) < 0 or g(v, v) = 0, and v ≠ 0. The causal character of a vector in Minkowski space is the property to be spacelike, timelike, or null (lightlike).
This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space E3. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space E3 is given as corollary. Moreover, our results are illustrated via some examples.
The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci’s theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work.
This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space E3 using the quasi-frame. In particular, we discuss some geometric properties and some characterizations of parameter curves of these surfaces in E3.
The modified orthogonal frame is an important tool to study analytic space curves whose curvatures have discrete zero points. In this article, by using the modified orthogonal frame, Mannheim curves and their partner curves are investigated in Minkowski 3-space. Some characterizations according to the curvatures and torsions of the curves are given. Finally, some relations under the conditions for Mannheim curves and their partner curves to be generalized helices are presented. All the possible cases for the partner curves to be spacelike and timelike are considered in the whole of the article.
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