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Keywords = Blaschke frame

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17 pages, 1309 KiB  
Article
Timelike Constant Axis Ruled Surface Family in Minkowski 3-Space
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2024, 16(6), 677; https://doi.org/10.3390/sym16060677 - 31 May 2024
Viewed by 469
Abstract
A timelike (TL) constant axis ruled surface in E13 (Minkowski 3-space), as determined by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In this article, we employ the symmetry through point geometry [...] Read more.
A timelike (TL) constant axis ruled surface in E13 (Minkowski 3-space), as determined by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In this article, we employ the symmetry through point geometry of Lorentzian dual curves and the line geometry of TL ruled surfaces. This produces the capability to expound a set of curvature functions that specify the local configurations of TL ruled surfaces. Then, we gain some new constant axis ruled surfaces in Lorentzian line space and their geometrical illustrations. Further, we also earn several organizations among a TL constant axis ruled surface and its striction curve. Full article
(This article belongs to the Section Mathematics)
►▼ Show Figures

Figure 1

Figure 1
<p>The hyperbolic and Lorentzian <math display="inline"><semantics> <mi mathvariant="script">DU</mi> </semantics></math> spheres.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold">e</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cosh</mo> <mover accent="true"> <mi>ω</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">z</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mo form="prefix">sinh</mo> <mover accent="true"> <mi>ω</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">z</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cylindroid.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>ϰ</mi> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>ϰ</mi> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> tangential surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 9
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> tangential surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 10
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> binormal surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 11
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> binormal surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 12
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cone.</p> Full article ">Figure 13
<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cylinder with <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">
16 pages, 545 KiB  
Article
Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space
by Sahar H. Nazra and Rashad A. Abdel-Baky
Axioms 2023, 12(7), 649; https://doi.org/10.3390/axioms12070649 - 29 Jun 2023
Cited by 3 | Viewed by 1012
Abstract
Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled [...] Read more.
Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
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Figure 1

Figure 1
<p>Archimedecs helicoid.</p> Full article ">Figure 2
<p>Right helicoid.</p> Full article ">Figure 3
<p>Hyperboloid of one-sheet.</p> Full article ">Figure 4
<p>A cone.</p> Full article ">Figure 5
<p>Helicoidal surface.</p> Full article ">Figure 6
<p>Bertrand offset of the helicoidal surface.</p> Full article ">Figure 7
<p>Helicoidal surface with its Bertrand offset.</p> Full article ">
16 pages, 713 KiB  
Article
Timelike Ruled Surfaces with Stationary Disteli-Axis
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2023, 15(5), 998; https://doi.org/10.3390/sym15050998 - 28 Apr 2023
Cited by 1 | Viewed by 1081
Abstract
This paper derives the declarations for timelike ruled surfaces with stationary timelike Disteli-axis by the E. Study map. This prepares the ability to determine a set of Lorentzian invariants which explain the local shape of timelike ruled surfaces. As a result, the Hamilton [...] Read more.
This paper derives the declarations for timelike ruled surfaces with stationary timelike Disteli-axis by the E. Study map. This prepares the ability to determine a set of Lorentzian invariants which explain the local shape of timelike ruled surfaces. As a result, the Hamilton and Mannhiem formulae of surfaces theory are attained at Lorentzian line space and their geometrical explanations are examined. Then, we define and explicate the kinematic geometry of a timelike Plűcker conoid created by the timelike Disteli-axis. Additionally, we provide the relationships through timelike ruled surface and the order of contact with its timelike Disteli-axis. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
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Figure 1

Figure 1
<p>The dual hyperbolic and dual Lorentzian unit spheres.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold">d</mi> <mo>^</mo> </mover> <mo>=</mo> <mo form="prefix">sinh</mo> <mover accent="true"> <mi>ϕ</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">ζ</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mo form="prefix">cosh</mo> <mover accent="true"> <mi>ϕ</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">ζ</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Timelike Plücker’s conoid.</p> Full article ">Figure 4
<p>General timelike helicoidal surface.</p> Full article ">Figure 5
<p>Lorentzian sphere.</p> Full article ">Figure 6
<p>Timelike Archimedes.</p> Full article ">Figure 7
<p>Timelike cone.</p> Full article ">Figure 8
<p>A timelike helicoid of the 1st kind.</p> Full article ">
11 pages, 468 KiB  
Article
Singularities of Non-Developable Ruled Surface with Space-like Ruling
by Rashad Abdel-Satar Abdel-Baky and Mohamed Khalifa Saad
Symmetry 2022, 14(4), 716; https://doi.org/10.3390/sym14040716 - 1 Apr 2022
Cited by 6 | Viewed by 2073
Abstract
Singularity theory is a significant field of modern mathematical research. The main goal in most problems of singularity theory is to understand the dependence of some objects in analysis and geometry, or physics; or from some other science on parameters. In this paper, [...] Read more.
Singularity theory is a significant field of modern mathematical research. The main goal in most problems of singularity theory is to understand the dependence of some objects in analysis and geometry, or physics; or from some other science on parameters. In this paper, we study the singularities of the spherical indicatrix and evolute of space-like ruled surface with space-like ruling. The main method takes advantage of the classical unfolding theorem in singularity theory, which is a classical method to study singularity problems in Euclidean space and Minkowski space. Finally, we provide an example to illustrate our results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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Figure 1

Figure 1
<p>The spacelike ruled surface <span class="html-italic">M</span>.</p> Full article ">Figure 2
<p>The evolute surface of <span class="html-italic">M</span>.</p> Full article ">Figure 3
<p>The space curve of the tangential surface.</p> Full article ">
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