scholarly journals The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Nural Yüksel
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Spacelike Curve ◽  
Bishop Frame ◽  
Straight Line ◽  
Base Curve ◽  
Distribution Parameters

We investigate the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. We obtain the distribution parameters, mean curvatures. We give some results and theorems related to be developable and minimal of them. Furthermore, we show that, if the base curve of the ruled surface is also an asymtotic curve and striction line, then the ruled surface is developable.

scholarly journals On the Blaschke approach of ruled surface

2003 ◽  
Vol 34 (2) ◽  
pp. 107-116 ◽  
Author(s):  
Rashad A. Abdel Baky
Keyword(s):  
Unit Sphere ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Necessary Condition ◽  
Original Surface ◽  
Distribution Parameters ◽  
Local Properties

In this paper using E. Study map and the Blaschke approach we studied a ruled surface as a curve on the dual unit sphere. The Blaschke approach proceeds by defining a sequence of ruled surfaces associated with the ruled surface. The relative positions of these surfaces and their distribution parameters characterize the local properties of the original surface. A necessary condition for a ruled surface to be closed is derived. Moreover, an example of application is investigated in detail.

On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories

1987 ◽  
Vol 109 (1) ◽  
pp. 101-106 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Differential Geometry ◽  
Ruled Surface ◽  
The Other ◽  
Ruled Surfaces ◽  
Vector Representation ◽  
Dual Vector ◽  
Vector Algebra ◽  
Distribution Parameters ◽  
Local Properties ◽  
Curvature Theory

The curvature theory of ruled surfaces has been studied in two different ways. The scalar formulation proceeds by defining a seqeunce of ruled surfaces associated with the trajectory ruled surface. The relative positions of these surfaces and their distribution parameters characterize the local properties of the original ruled surface. The other formulation uses dual vector algebra to transform the differential geometry of ruled surfaces into that of spherical curves. In each theory functions are obtained which characterize the shape of the ruled surface. This paper unites these formulations by deriving formulas relating the scalar and dual curvature functions. This provides the ability to compute either set of curvature properties from either the scalar or dual vector representation of the ruled surface. The ruled surface generated by a line fixed in a body undergoing a screw displacement is examined in detail.

scholarly journals Slant ruled surfaces and slant developable surfaces of spacelike curves in Lorentz-Minkowski 3-space

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4875-4895
Author(s):  
Handan Yıldırım
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Frenet Frame ◽  
Spacelike Curve ◽  
Developable Surfaces

In this paper, by means of the Lorentzian Frenet frame along a spacelike curve in Lorentz-Minkowski 3-space, we construct slant ruled surfaces and slant developable surfaces with different director curves which belong to one-parameter families of the pseudo-spheres in this space. Moreover, for each slant ruled surface with each director curve, we search if this slant ruled surface has any singularities or not. Furthermore, for the cases in which the singularities appear, we determine the singularities of non-lightlike and non-cylindrical slant developable surfaces and also investigate the singularities of slant ruled surfaces.

Non-lightlike ruled surfaces with constant curvatures in Minkowski 3-space

2018 ◽  
Vol 15 (04) ◽  
pp. 1850068 ◽  
Author(s):  
Ahmad Tawfik Ali
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surface ◽  
Timelike Curve ◽  
Ruled Surfaces ◽  
Base Curve ◽  
Timelike Ruled Surface ◽  
Interesting Theorem

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] are the tangent, principal normal and binormal vectors of an arbitrary timelike curve [Formula: see text]. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.

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

2020 ◽  
Vol 14 (2) ◽  
pp. 11
Author(s):  
E. M. Solouma ◽  
M. M. Wageeda ◽  
M. A. Soliman ◽  
M. Bary
Keyword(s):  
Minkowski Space ◽  
Space Time ◽  
Three Dimension ◽  
Geometric Properties ◽  
Spacelike Curve ◽  
Bishop Frame ◽  
Minkowski Space Time ◽  
Base Curve ◽  
Special Case

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.

scholarly journals Characteristics of the double-cycled motion-ruled surface of the Schatz linkage based on differential geometry

2014 ◽  
Vol 229 (5) ◽  
pp. 957-964 ◽  
Author(s):  
Lei Cui ◽  
Jian S Dai ◽  
Chung-Ching Lee
Keyword(s):  
Differential Geometry ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Full Rotation ◽  
Kinematic Properties

This paper applies Euclidean invariants from differential geometry to kinematic properties of the ruled surfaces generated by the coupler link and the constraint-screw axes. Starting from investigating the assembly configuration, the work reveals two cycle phases of the coupler link when the input link finishes a full rotation. This leads to analysis of the motion ruled surface generated by the directrix along the coupler link, where Euclidean invariants are obtained and singularities are identified. This work further presents the constraint ruled surface that is generated by the constraint screw axes and unveils its intrinsic characteristics.

scholarly journals On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Cumali Ekici ◽  
Yasin Ünlütürk ◽  
Mustafa Dede ◽  
B. S. Ryuh
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
End Effector ◽  
Timelike Ruled Surface ◽  
Curvature Theory ◽  
Differential Properties

The trajectory of a robot end-effector is described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way, the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The curvature theory of a ruled surface generated by a line fixed in the end-effector referred to as the tool line is used for more accurate motion of a robot end-effector. In the present paper, we first defined tool trihedron in which tool line is contained for timelike ruled surface with timelike ruling, and transition relations among surface trihedron: tool trihedron, generator trihedron, natural trihedron, and Darboux vectors for each trihedron, were found. Then differential properties of robot end-effector's motion were obtained by using the curvature theory of timelike ruled surfaces with timelike ruling.

scholarly journals Kähler Yamabe minimizers on minimal ruled surfaces

2002 ◽  
Vol 90 (2) ◽  
pp. 180
Author(s):  
Christina W. Tønnesen-Friedman
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Ruled Surface ◽  
Ruled Surfaces

It is shown that if a minimal ruled surface $\mathrm{P}(E) \rightarrow \Sigma$ admits a Kähler Yamabe minimizer, then this metric is generalized Kähler-Einstein and the holomorphic vector bundle $E$ is quasi-stable.

On the Biais Passé

2016 ◽  
pp. 337-366
Author(s):  
João Pedro Xavier ◽  
Eliana Manuel Pinho
Keyword(s):  
String Model ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Descriptive Geometry ◽  
13Th Century ◽  
Aesthetic Value ◽  
Structural Efficiency ◽  
Stone Cutting ◽  
String Models

Among the famous dynamic string models conceived by Théodore Olivier (1793-1853) as a primary didactic tool to teach Descriptive Geometry, there are some that were strictly related to classic problems of stereotomy. This is the case of the biais passé, which was both a clear illustration of a special warped ruled surface and an example of how constructors dealt with the problem of building a skew arch, solving structural and practical stone cutting demands. The representation of the biais passé in Olivier's model achieved a perfect correspondence to its épure with Monge's Descriptive Geometry. This follow from the long development of representational tools, since the 13th century sketch of an oblique passage, as well as the improvement of constructive procedures for skew arches. Paradoxically, when Olivier presented his string model, the importance of the biais passé was already declining. Meanwhile other ruled surfaces were appropriated by architecture, some of which acquiring, beyond their inherent structural efficiency, a relevant aesthetic value.

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