Dual Lorentzian Space Research Articles

Associated curves of a Frenet curve in the dual Lorentzian space

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 .

Characterizations of dual curves and dual focal curves in dual Lorentzian spaceD31

In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dual Lorentzian space $ D_{1}^{3} .$ We have studied a dual curve in different situations in dual Lorentzian space $D^{3}_{1} $ and have found Bishop Darboux vector and some relations according to this vector field, Bishop frame and focal curve of the present dual curve. It has been shown that Bishop Darboux vector has a similar amount in three different cases of a dual curve and the first dual focal curvature of the aforementioned curve is constant function.

The relations between null geodesic curves and timelike ruled surfaces in dual Lorentzian space $\mathbb{D}_{1}^{3}$

The relations between null geodesic curves and timelike ruled surfaces in dual Lorentzian space $\mathbb{D}_{1}^{3}$

On closed timelike and spacelike ruled surfaces

In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non‐lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.

SLANT HELICES IN DUAL LORENTZIAN SPACE D3 1

In this paper, we consider a unit speed dual Lorentzian curve a in dual Lorentzian space D31 and denote by {T ,N, B } the dual Frenet frame of a. We say that a is a slant helix if there exists a non-zero dual constant vector eld U in D 3 1 such that the dual function is a dual constant. Moreover, we give some characterizations of slant helice in terms of their dual curvatures. Finally, we show that dual tangent indicatrices and dual binormal indicatrices of slant helices are dual helices.

On the kinematic interpretation of timelike ruled surfaces

Timelike ruled surfaces are studied in dual Lorentzian space [Formula: see text] by considering E. Study Mapping and Blaschke frame. A reference timelike ruled surface is considered and associated surfaces are defined. First, it is shown that the surface generated by the instantaneous screw axis (ISA) is a Mannheim offset of reference surface. Later, the kinematic interpretations between these surfaces are introduced by means of Blaschke invariants.

On the invariants of Mannheim offsets of timelike ruled surfaces with spacelike rulings

In this paper, we give the characterizations for Mannheim offsets of timelike ruled surfaces with spacelike rulings in dual Lorentzian space [Formula: see text]. We obtain the relations between terms of their integral invariants and also we give new characterization for the Mannheim offsets of developable timelike ruled surface. Moreover, we find relations between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.

Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces

In this paper, we define dual Darboux frame of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Moreover, we obtain some conditions for Mannheim offsets of spacelike ruled surfaces to be developable.

On dual r-elastic line

In this paper, we derive intrinsic equations for dual r-elastic line deformed on non-null surface by external field in dual Lorentzian space.

Curvature Motion on Dual Hyperbolic Unit Sphere H<sup>2</sup><sub style="margin-left:-8px;">0</sub>

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere H20 in the dual Lorentzian space D31 with dual signature (+,+-) . We carry the obtained results to the Lorentzian line space R31 by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in R31.

On Some Characterizations of Ruled Surface of a Closed Timelike Curve in Dual Lorentzian Space

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed curve in dual Lorentzian space.

On instantaneous invariants in dual Lorentzian space kinematics

In this work, we study the concepts of canonical systems and instantaneous invariants in dual Lorentzian space. It gives a systematic approximation about determining the instantaneous invariants. Important for the characterization of higher-order intrinsic properties of a ruled surface, the instantaneous invariants are derived on the basis of line coordinates.

A STUDY ON RECTIFYING CURVES IN THE DUAL LORENTZIAN SPACE

In this work, we give some characterizations of rectifying curves in dual Lorentzian space. Also, we show that rectifying dual Lorentzian curves can be stated by the aid of dual unit spherical curves.

Dual Lorentzian spherical motions and dual Euler–Savary formula

Abstract In this paper, defining one-parameter dual Lorentzian motion in three-dimensional dual Lorentzian space D 1 3 and we have obtained the relations between the absolute, relative and sliding velocities of these motions. In addition we have given the interconnections between one-parameter dual Lorentzian motion in three-dimensional dual Lorentzian space D 1 3 and one-parameter Lorentzian motion in three-dimensional Lorentzian space, IR 1 3 . Furthermore, the correlations between the fixed and the moving dual pole curves of one-parameter dual Lorentzian motion have been obtained. At the end of the study dual Euler–Savary formula for one-parameter dual Lorentzian motion has been given.

Elliptic motion on dual hyperbolic unit sphere

Planar and spherical elliptic motions had been introduced by Karger and Novák [A. Karger, J. Novák, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, London, 1985]. Also, a dual spherical elliptic motion has been defined by Yapar [Z. Yapar, Spatial conchoidal and elliptic motions, Mechanism and Machine Theory 27 (1) (1992) 75–91]. In this paper, we define the elliptic motion on a dual hyperbolic unit sphere H ∼ 0 2 in the dual Lorentzian space D 1 3 with dual signature (+, +, −) and carry the results to the Lorentzian line space IR 1 3 by means of Study’s mapping.

The Sine and Cosine Rules for Pure Triangles on the Dual Lorentzian Unit Sphere \({{{\overset{\sim}{S}}_{1}}^{2} }\)

In this work, we proved the sine and cosine rules for a spherical pure triangle on the dual Lorentzian unit sphere S ˜ 1 2 in the dual Lorentzian space D13.

The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres $H_{0}^{2}$ and $S_{1}^{2}$

H. H. Hacisalihoğlu ('Study map of a circle', "Journal of the Faculty of Science of Karadeniz Technical University" 1 (7) (1977), 69-80) shows that the E. Study map of a circle on a dual unit sphere S² is a family of hyperboloids of one sheet with two parameters. In this paper we calculate and discuss the E. Study maps of circles which lie on the dual hyperbolic and Lorentzian unit spheres $H_{0}^{2}$ and $S_{1}^{2}$ at the dual Lorentzian space $D_{1}^{3}$. Furthermore, we examine some special cases, each of which is a geometrical result.

The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres <i>H</i><sup>2</sup><sub>0</sub> and <i>S</i><sup>2</sup><sub>1</sub>

H.H. Hacisaliho-lu ('Study map of a circle' Journal of the Faculty of Science of Karadeniz Technical University 1 (7) (1977), 69-80) shows that the E. Study map of a circle on a dual unit sphere s2 is a family of hyperboloids of one sheet with two parameters. In this paper we calculate and discuss the E. Study maps of circles which lie on the dual hyperbolic and Lorentzian unit spheres Ho and S? at the dual Lorentzian space D3. Furthermore, we examine some special cases, each of which is a geometrical result.

The Cosine Hyperbolic and Sine Hyperbolic Rules for Dual Hyperbolic Spherical Trigonometry

The dual hyperbolic unit sphere \(H_{0}^{2}\) is the set of all dual time-like units vectors in the dual Lorentzian space \(D_{1}^{3}\) with signature (+,+,-). In this paper, we give the cosine hyperbolic and sine hyperbolic-rules for a dual dual hyperbolic spherical triangle \(\tilde{A}\tilde{B}\tilde{C}\) which its sides are great-circle-arcs.