A study on timelike circular surfaces in Minkowski 3-space

2020 ◽  
Vol 17 (06) ◽  
pp. 2050074
Author(s):  
Rashad A. Abdel-Baky ◽  
Nadia Alluhaibi ◽  
Akram Ali ◽  
Fatemah Mofarreh
Keyword(s):  
Geodesic Curvature ◽  
Parameter Family ◽  
Lines Of Curvature ◽  
Constant Radius ◽  
Spherical Indicatrix ◽  
New Type ◽  
Circular Surface ◽  
Fixed Radius

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.

Kinematic Geometry of Circular Surfaces With a Fixed Radius Based on Euclidean Invariants

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
Lei Cui ◽  
Delun Wang ◽  
Jian S. Dai
Keyword(s):  
Mechanism Analysis ◽  
Necessary And Sufficient Condition ◽  
Principal Curvatures ◽  
Lines Of Curvature ◽  
Kinematic Geometry ◽  
Circular Surface ◽  
Fixed Radius ◽  
Necessary And Sufficient ◽  
First Time ◽  
The Relationship

A circular surface with a fixed radius can be swept out by moving a circle with its center following a curve, which acts as the spine curve. Based on a system of Euclidean invariants, the paper identifies those circular surfaces taking lines of curvature as generating circles and further explores the properties of the principal curvatures and Gaussian curvature of the tangent circular surfaces. The paper then applies the study to mechanism analysis by proving the necessary and sufficient condition for a circular surface to be generated by a serially connected C′R, HR, or RR mechanism, where C′ joint can be visualized as a special H joint with a variable pitch of one degree of freedom. Following the analysis, this paper reveals for the first time the relationship between the invariants of a circular surface and the commonly used D-H parameters of C′R, HR, and RR mechanisms.

A Euclidean Invariants Based Study of Circular Surfaces With Fixed Radius

2007 ◽  
Author(s):  
Lei Cui ◽  
Delun Wang
Keyword(s):  
Complete System ◽  
Distribution Parameter ◽  
Moving Frame ◽  
Necessary Condition ◽  
Spherical Indicatrix ◽  
Circular Surface ◽  
Fixed Radius ◽  
Curvature And Torsion ◽  
The Relationship ◽  
Sufficient And Necessary Condition

In this paper a complete system of Euclidean invariants is presented to study circular surfaces with fixed radius. The study of circular surfaces is simplified to the study of two curves: the spherical indicatrix of the unit normals of circle planes and the spine curve. After the geometric meanings of these Euclidean invariants are explained, the distribution parameter of a circular surface is defined. If the value of the distribution parameter of a circular surface is 0, the circular surface is a sphere. Then the relationship between the moving frame {E1, E2, E3} and the Frenet frame {t, n, b} of the spine curve is investigated, and the expressions of the curvature and torsion of the spine curve are obtained based on these Euclidean invariants. The fundamental theorem of circular surfaces is first proved. Next the first and second fundamental forms of circular surfaces are computed. The last part of this paper is devoted to constraint circular surfaces. The sufficient and necessary condition for a general circular surface to be one that can be generated by a series-connected C’R, HR, RR, or PR mechanism is proved.

scholarly journals The Development of Log Aesthetic Patch and Its Projection onto the Plane

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 160
Author(s):  
Yee Meng Teh ◽  
R. U. Gobithaasan ◽  
Kenjiro T. Miura ◽  
Diya’ J. Albayari ◽  
Wen Eng Ong
Keyword(s):  
Outer Part ◽  
Geodesic Curvature ◽  
Surface Patch ◽  
Hyperbolic Paraboloid ◽  
Shipbuilding Industry ◽  
First Derivative ◽  
Cold Bending ◽  
Lines Of Curvature ◽  
New Type ◽  
Boundary Curves

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.

scholarly journals Mannheim Offsets of Ruled Surfaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Keziban Orbay ◽  
Emin Kasap ◽  
İsmail Aydemir
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Geodesic Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Arc Length ◽  
Spherical Indicatrix ◽  
Constant Distance ◽  
Mannheim Offset ◽  
Three Dimensional Space

In a recent works Liu and Wang (2008; 2007) study the Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it. Moreover, we obtain that the Mannheim offset of developable ruled surface is constant distance from it. Finally, examples are also given.

scholarly journals One parameter family of b-m₁ developable surfaces of biharmonic new type b-slant helices according to Bishop frame in the sol space Sol³

2013 ◽  
Vol 31 (2) ◽  
pp. 121
Author(s):  
Mahmut Ergüt ◽  
Talat Körpınar ◽  
Essin Turhan
Keyword(s):  
Parameter Family ◽  
Type B ◽  
Developable Surfaces ◽  
Bishop Frame ◽  
Slant Helix ◽  
New Type ◽  
Inextensible Flows ◽  
Sol Space

In this paper, we study inextensible flows of b-m₁ developable surfaces of biharmonic new type b-slant helix in the Sol³. We characterize one parameter family of the b-m₁ developable surfaces in terms of their Bishop curvatures.

Directional spherical indicatrices of timelike space curve

2020 ◽  
Vol 17 (11) ◽  
pp. 2030004
Author(s):  
Gul Ugur Kaymanli ◽  
Mustafa Dede ◽  
Cumali Ekici
Keyword(s):  
Geodesic Curvature ◽  
Space Curve ◽  
Slant Helix ◽  
Spherical Indicatrix

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

A CHARACTERIZATION OF THE CATENOID AND HELICOID

2013 ◽  
Vol 24 (06) ◽  
pp. 1350045 ◽  
Author(s):  
CARLOS M. C. RIVEROS ◽  
ARMANDO M. V. CORRO
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Geodesic Curvature ◽  
Lines Of Curvature ◽  
Asymptotic Lines

In this paper we show that a connected non-planar minimal surface whose asymptotic lines have the same geodesic curvature up to sign is a catenoid. As an application of this result we show that a connected non-planar minimal surface whose lines of curvature have the same geodesic curvature up to sign is a helicoid. Moreover, we show that the coordinates curves of the associate minimal surfaces to catenoid have the same geodesic curvature up to sign.

scholarly journals Isometric bending requires local constraints on free edges

2019 ◽  
Vol 24 (12) ◽  
pp. 4051-4077 ◽  
Author(s):  
Jemal Guven ◽  
Martin Michael Müller ◽  
Pablo Vázquez-Montejo
Keyword(s):  
Boundary Conditions ◽  
Thin Sheet ◽  
Geodesic Curvature ◽  
Outer Edge ◽  
Radial Compression ◽  
Arc Length ◽  
Lateral Tension ◽  
Deficit Angle ◽  
Fixed Radius ◽  
Free Edges

While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naïvely one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manifestly unphysical constraints are met by the boundary geometry. In this article, we identify the source of the problem: not only the local arc-length but also the geodesic curvature need to be constrained explicitly on all free edges. We derive the boundary conditions which follow. In contrast to conventional wisdom, there is no need to introduce boundary layers. This framework is applied to isolated conical defects, both with deficit as well, but more briefly, as surplus angles. Using these boundary conditions, we show that the lateral tension within a circular cone of fixed radius is equal but opposite to the radial compression, and independent of the deficit angle itself. We proceed to examine the effect of an oblique outer edge on this cone perturbatively demonstrating that both the correction to the geometry as well as the stress distribution in the cone kicks in at second order in the eccentricity of the edge.

Ribaucour surfaces of harmonic type

2021 ◽  
Author(s):  
Armando M. V. Corro ◽  
Carlos M. C. Riveros ◽  
Karoline V. Fernandes
Keyword(s):  
Holomorphic Functions ◽  
Parameter Family ◽  
Isolated Singularities ◽  
Type Representation ◽  
Lines Of Curvature ◽  
Weierstrass Type Representation

We introduce the class of Ribaucour surfaces of harmonic type (in short HR-surfaces) that generalizes the Ribaucour surfaces related to a problem posed by Élie Cartan. We obtain a Weierstrass-type representation for these surfaces which depends on three holomorphic functions. As application, we classify the HR-surfaces of rotation, present examples of complete HR-surfaces of rotation with at most two isolated singularities and an example of a complete HR-surface of rotation with one catenoid type end and one planar end. Also, we present a 5-parameter family of cyclic HR-surfaces foliated by circles in non-parallel planes. Moreover, we classify the isothermic HR-surfaces with planar lines of curvature.

Externally Pressurized Two Layer Film Bearing—Gas Bearing With Liquid Layer

1981 ◽  
Vol 103 (4) ◽  
pp. 573-577
Author(s):  
T. Nakahara ◽  
T. Kurisu ◽  
H. Aoki
Keyword(s):  
Liquid Film ◽  
Liquid Layer ◽  
Thrust Bearing ◽  
Load Capacity ◽  
Thin Liquid Film ◽  
Gas Flows ◽  
Constant Radius ◽  
New Type ◽  
Film Stiffness ◽  
Gas Bearing

A new type of gas lubricated externally pressurized thrust bearing was developed. The bearing consists of a central hole for feeding the gas and a row of 24 small holes located circumferencially at a constant radius for supplying a small amount of liquid which, by the combined wetting action and the gas flows, forms a thin liquid layer adhering onto one of the surfaces. The pattern of the liquid film was observed experimentally, and the bearing performance influenced by the thin liquid film was determined. The load capacity and film stiffness were found to increase approximately 2 and 6 times respectively.

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