Geometric Design of Smooth Composite Ruled Surface Strips Using Dual Spherical Geometry

1995 ◽  
Author(s):  
Q. J. Ge ◽  
Donglai Kang
Keyword(s):  
Spherical Geometry ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Coordinate Frame ◽  
Geometric Continuity ◽  
Dual Numbers ◽  
Parametric Curve ◽  
Geometric Conditions ◽  
Line Segmentation ◽  
Unit Vectors

Abstract This paper deals with geometric construction of smooth composite ruled surface strips. Oriented lines that constitute the rulings of the ruled surfaces are represented by unit vectors with three components over the ring of dual numbers. The problem of designing a smooth ruled surface is studied as that of designing a one-real-parametric curve on the unit dual sphere. Geometric conditions for piecing two ruled surfaces smoothly are developed using differential geometry of curves on the dual sphere. A coordinate-frame invariant method for line segmentation is also presented. Finally, a geometric algorithm is presented for constructing composite Bézier ruled surface strips with second-order geometric continuity. The resulting surface strips are coordinate-frame invariant and their rulings are more uniformly parameterized than those obtained with other methods.

General Principles for Formation of Ruled Surfaces. Part 3

2019 ◽  
Vol 7 (2) ◽  
pp. 13-27
Author(s):  
Николай Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Descriptive Geometry ◽  
Guide Line ◽  
Geometric Conditions ◽  
Single Method ◽  
The Law ◽  
Straight Lines ◽  
Guide Lines

We continue to consider the formation of ruled surfaces with a single method of their formation. In the first and second parts have been introduced more than forty options for specifying surfaces. These formations with the help of guide lines and surfaces are considered in a new aspect – as formation in science and production of all of ruled surfaces. In this paper, we consider new options for specifying ruled surfaces. Generalized the task for torso surface. If in textbooks on descriptive geometry torso surface is given as 1∞ straight lines, tangent to the spatial line, the proposed version of the are considered three guides: two curves (surface) plus a plane touching to both curves (surfaces). It is shown that three guides are also required to set screw ruled surfaces. The concept of a limit ruled surface is introduced to determine the region of existence of ruled surfaces. The table of the simplest geometrical figures for obtaining congruences is given. A number of examples of congruences obtained by using two guides are given. All these examples once again confirmed the validity of the law of assignment of ruled surfaces using three guides and three geometric conditions characterizing the ratio of the forming line to these three guides. The three geometric conditions are the contact of the forming line to the guide surface and the intersection of the forming line with the guide line. The proposed task of ruled surfaces can be used in the consideration of ruled surfaces in lectures on descriptive geometry and other geometric disciplines.

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.

The orientational average of exp[2πi(h.x + k.y + l.z)]

1978 ◽  
Vol 34 (6) ◽  
pp. 955-959 ◽  
Author(s):  
J. Brosius
Keyword(s):  
Molecular Structure ◽  
A Priori ◽  
Coordinate Frame ◽  
Phase Problem ◽  
Fixed Frame ◽  
Orthogonal Frame ◽  
Unit Vectors

The unit vectors e l, e 2, e 3 form a fixed orthogonal coordinate frame. The unit vectors e l ', e 2 ', e 3 ' form a movable orthogonal frame with the same origin as the fixed frame. All orientations in space of the movable frame are assumed to be equally probable. Under these conditions the average of exp [2πi(Σ3 i,j = 1a ij e i .e ' j )] is calculated. As an application the average of exp [2πi(h.x + k.y + l.z)] is calculated where the vectors h,k,l are specified and where the magnitudes of the vectors x,y,z and the angles between them are specified. This integral is of importance in utilizing a priori knowledge of molecular structure as an aid in solving the phase problem.

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.

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