Minkowski 3-space Research Articles

Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme.

Some characterizations of timelike rectifying curves in de sitter 3 space

De Sitter space is a non-flat Lorentzian space form with positive constant curvature which plays an important role in the theory of relativity. In this paper, we define the notions of timelike rectifying curve and timelike conical surface in De Sitter 3-space as Lorentzian viewpoint. Moreover, we give some nice characterizations and results of a timelike rectifying curves with respect to curve-hypersurface frame in De Sitter 3-space which is a three dimensional pseudo-sphere in Minkowski 4-space.

Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in E_{1}^{3}

Equiform geometry is considered as a generalization of the other geometries. In this paper, involute and evolute curves are studied in the case of the curve α is an equiform spacelike with a timelike equiform principal normal vector N. Furthermore, the equiform frames of the involute and evolute curves are obtained. Also, the equiform curvatures of the involute and evolute curves are obtained in Minkowski 3-space.

Construction of Minkowski Pythagorean hodograph B-spline curves

Following and extending the recent results of Albrecht et al. (2017) for planar Pythagorean hodograph (PH) B-spline curves to the Minkowski 3-space, we introduce a class of Minkowski Pythagorean hodograph (MPH) B-spline curves. The distinguished property of these curves is that the Minkowski norm of their hodograph is a B-spline function. We focus mainly on the clamped case and using Clifford algebra representation we present formulas for their construction. The closed case is also mentioned. Then we solve two practical problems – construction of MPH B-spline curves with control polygon close to a given control polygon, and construction of MPH B-spline curves going through given points. We emphasize symbolic solutions wherever it is possible. The results and approaches are illustrated on several examples.

Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space

In this paper, we define a relativeMinkowski normal plane and relative tangent vector TM:We construct a new relative S-frame (Shnoda-Saad frame) of regular spacelike curves on timelike surfaces. It depends only on the curve lies on the surface, Euclidean and Minkowski unit normal vectors. Also, we define S-curve according to this frame with some related theorems.

Mixed-type curves and the lightcone frame in Minkowski 3-space

In this paper, we study mixed-type curves in Minkowski 3-space. Mixed-type curves are regular curves, and there are both non-lightlike points and lightlike points in a mixed-type curve. For non-lightlike curves and null curves in Minkowski 3-space, we can study them by a Frenet frame or a Cartan frame, respectively. But for mixed-type curves, the two frames will not work. As far as we know, no one has yet given a frame to study them in Minkowski 3-space. So, we give the lightcone frame in order to provide a tool for studying this type curves in mathematical and physical research. As an application of the lightcone frame, we define an evolute of a mixed-type curve. We also give some examples to show the evolutes.

A study on timelike circular surfaces in Minkowski 3-space

This paper studies a smooth one-parameter family of standard Lorentzian circles with fixed radius. Such a surface is called a timelike circular surface with constant radius. We call each circle a generating circle. A new type of timelike circular surfaces was identified and coined as the timelike tangent circular surface. The new timelike tangent circular surface has the property of all generating circles being lines of curvature and its Gaussian and mean curvatures being independent of the geodesic curvature of the spherical indicatrix.

A NEW CHARACTERIZATION OF CURVES IN MINKOWSKI 4-SPACE E_1^4

In this study, we attend to the curves whose position vectors are written as alinear combination of their Serret-Frenet vectors in Minkowski 4-Space E_1^4. We characterize such curves with regard to their curvatures. Further, we get certain consequences of T-constant and N-constant types of curves in E_1^4.

On Null Curves in Minkowski 3-Space and Its Fractal Folding

In this paper, a form for Frenet equations of all null curves in Minkowski 3-space has been presented. New types of foldings of curves are obtained. The connection between folding, deformation and Frenet equations of curves are also deduced.

Siacci's Theorem for Frenet Curves in Minkowski 3-Space

For motion of a material point along a space curve, due to Siacci [1], a resolution of the acceleration vector is well known. In this resolution, the acceleration vector is stated as the sum of two special oblique components in the osculating plane to the curve. In this paper, we have studied the Siacci’s theorem for non-relativistic particles moving along the Frenet curves at very low speeds relative to the speed of light in Minkowski 3-space. Moreover, an illustrative example is given to show how the aforesaid theorem works. This theorem is a new contribution to the field and it may be useful for some specific applications in mathematical and computational physics.

Darboux Associated Curves of a Null Curve on Pseudo-Riemannian Space Forms

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained.

Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space

Abstract The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space 𝐄 1 3 {\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ℒ 2 {\mathcal{L}^{2}} ). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} . We shall discuss the causal character of the second derivative of unit speed spacelike curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.

Geometric Properties of Special Spacelike Curves in Three-Dimension Minkowski Space-Time

In this paper, we introduce a special spacelike Smarandache curves  reference to the Bishop frame of a regular spacelike curve  in Minkowski 3-space . From that point, we investigate the Frenet invariants of a special case in  and we obtain some properties of these curves when the base curve  is contained in a plane. Lastly, we shall give two examples to illustrate these curves.

Elastic strips with null and pseudo-null directrix

The solution of the variational problem which gives elastic strips in Minkowski [Formula: see text]-space is separately obtained for elastic strips with null and pseudo-null directrix. First, critical points of the modified Sadowsky functional (which depends on the modified torsion) for elastic strips with null directrix are characterized by three Euler–Lagrange equations. A connection is established between elastic curves on the two-dimensional null cone and elastic strips with null directrix. Then conservation laws of elastic strips with null directrix in Minkowski 3-space are given. Second, equilibrium equations for elastic strips with pseudo-null directrix are determined and solved. It is also shown the tangent and the binormal of a critical curve of the Sadowsky functional correspond to a null elastic curve in de Sitter 2-space and a spacelike elastic curve in the two-dimensional null cone, respectively. Finally, two conservation laws for elastic strips with pseudo-null directrix are derived.

Structure Functions of Pseudo Null Curves in Minkowski 3-Space

In this work, the embankment surfaces with pseudo null base curves are investigated in Minkowski 3-space. The representation formula of pseudo null curves is obtained via the defined structure functions and the k-type pseudo null helices are discussed completely. Based on the theories of pseudo null curves, a class of embankment surfaces are constructed and characterized by the structure functions of the pseudo null base curves.

A New Angular Measurement in Minkowski 3-Space

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

Natural Mates of Non-Null Frenet Curves in Minkowski 3-Space

In this paper, we give the definition of the natural mate of a non-null Frenet curve in Minkowski 3-spaces. The main purpose of this paper is to prove some relationships between a non-null Frenet curve and its natural mate. In particular, we obtain some necessary and suffcient conditions for the natural mate of a non-null Frenet curve to be a slant helix, a spherical curve, or a curve of constant curvature. Several applications of our main results are also presented.

Special Curves According to Bishop Frame in Minkowski 3-Space

Abstract Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space E 1 3 [E_1^3 . We define helix and slant helix according to Bishop frame in E 1 3 [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

Construction of the spacelike constant angle surface family in Minkowski 3-space

A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.

The characterizations of some special Frenet curves in Minkowski 3-space

The characterizations of some special Frenet curves in Minkowski 3-space