scholarly journals Parallel surfaces to normal ruled surfaces of general helices in the sol space Sol³

2013 ◽  
Vol 31 (2) ◽  
pp. 245 ◽  
Author(s):  
Talat Körpinar ◽  
Essin Turhan
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Parallel Surfaces ◽  
Sol Space

In this paper, parallel surfaces to normal ruled surface of general helices in the Sol³ are studied. Also, explicit parametric equations of parallel surfaces to normal ruled surface of general helices in the Sol³ are found.

A new characterization of null scrolls as parallel surfaces

2021 ◽  
Author(s):  
Yasin Ünlütürk
Keyword(s):  
Vector Fields ◽  
Ruled Surface ◽  
Point Of View ◽  
Ruled Surfaces ◽  
Parallel Surface ◽  
Parallel Surfaces

In this paper, we present a theorem which explains the necessary onditions for parallel surfaces of null scrolls to be surfaces again. Then, we give some properties of parallel ruled surfaces such as the frame fields in terms of the original base curve’s frame vectors, the cases of cross products of these frames’ vector fields. Also, we show the parallel ruled surfaces to be developable from a different point of view. Additionally, we obtain a relation between Gauss curvatures and mean curvatures. Finally, we characterize the striction curve of the parallel ruled surface. In the example, we draw a graph for a particular null scroll and its parallel surface.

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.

Bisecant curves of ruled surfaces

1933 ◽  
Vol 29 (3) ◽  
pp. 382-388
Author(s):  
W. G. Welchman
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Normal Space ◽  
Double Points ◽  
Image Position

The bisecant curves of a ruled surface, that is to say the curves on the surface which meet each generator in two points, are fundamental in the consideration of the normal space of the ruled surface. It is well known that if is a bisecant curve of order ν and genus π on a ruled surface of order N and genus P, thenprovided that the curve has no double points which count twice as intersections of a generator of the ruled surface.

scholarly journals On the second Gaussian curvature of ruled surfaces in Euclidean $3$-space

2006 ◽  
Vol 37 (3) ◽  
pp. 221-226 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Mean ◽  
Second Gaussian Curvature

In this paper, we mainly investigate non developable ruled surface in a 3-dimensional Euclidean space satisfying the equation $K_{II} = KH$ along each ruling, where $K$ is the Gaussian curvature, $H$ is the mean curvature and $K_{II}$ is the second Gaussian curvature.

scholarly journals Generalized normal ruled surface of a curve in the Euclidean 3-space

2021 ◽  
Vol 13 (1) ◽  
pp. 217-238
Author(s):  
Onur Kaya ◽  
Mehmet Önder
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Asymptotic Curves ◽  
Lines Of Curvature

Abstract In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.

Some New Theorems in Geometry of a Surface

1926 ◽  
Vol 13 (180) ◽  
pp. 1-6 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Focal Surface ◽  
Family Of Curves

The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

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