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Search Results (4)

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Keywords = Dual Lorentzian space

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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
Cited by 1 | Viewed by 735
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)
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14 pages, 1514 KiB  
Article
Kinematic Geometry of Timelike Ruled Surfaces in Minkowski 3-Space E13
by Nadia Alluhaibi and Rashad A. Abdel-Baky
Symmetry 2022, 14(4), 749; https://doi.org/10.3390/sym14040749 - 5 Apr 2022
Cited by 10 | Viewed by 2473
Abstract
Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry, most notably line geometry, and in other closely related areas. In this study, we take advantage of the symmetry properties of both dual [...] Read more.
Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry, most notably line geometry, and in other closely related areas. In this study, we take advantage of the symmetry properties of both dual space and original space in order to transfer problems in original space to dual space. We use E. Study Mappingas a direct method for analyzing the kinematic geometry of timelike ruled and developable surfaces. Then, the invariants for a spacelike line trajectory are studied and the well-known formulae of Hamilton and Mannheim on the theory of surfaces are provenfor the line space. Meanwhile, a timelike Plücker conoid generated by the Disteli-axis is derived and its kinematic geometry is discussed. Finally, some equations for particular timelike ruled surfaces, such as the general timelike helicoid, the Lorentzian sphere, and the timelike cone, are derived and plotted. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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9 pages, 215 KiB  
Article
The Sine and Cosine Rules for Pure Triangles on the Dual Lorentzian Unit Sphere \({{{\overset{\sim}{S}}_{1}}^{2} }\)
by M. Kazaz
Math. Comput. Appl. 2005, 10(2), 193-201; https://doi.org/10.3390/mca10020193 - 1 Aug 2005
Viewed by 1269
Abstract
In this work, we proved the sine and cosine rules for a spherical pure triangle on the dual Lorentzian unit sphere S ˜ 1 2 in the dual Lorentzian space D13. [...] Read more.
In this work, we proved the sine and cosine rules for a spherical pure triangle on the dual Lorentzian unit sphere S ˜ 1 2 in the dual Lorentzian space D13. Full article
6 pages, 348 KiB  
Article
The Cosine Hyperbolic and Sine Hyperbolic Rules for Dual Hyperbolic Spherical Trigonometry
by H. H. UĞURLU and H. GÜNDOĞAN
Math. Comput. Appl. 2000, 5(3), 185-190; https://doi.org/10.3390/mca5020185 - 1 Dec 2000
Viewed by 1416
Abstract
The dual hyperbolic unit sphere \(H_{0}^{2}\) is the set of all dual time-like units vectors in the dual Lorentzian space \(D_{1}^{3}\) with signature (+,+,-). In this paper, we give the cosine hyperbolic and sine hyperbolic-rules for a dual dual hyperbolic spherical triangle \(\tilde{A}\tilde{B}\tilde{C}\) [...] Read more.
The dual hyperbolic unit sphere \(H_{0}^{2}\) is the set of all dual time-like units vectors in the dual Lorentzian space \(D_{1}^{3}\) with signature (+,+,-). In this paper, we give the cosine hyperbolic and sine hyperbolic-rules for a dual dual hyperbolic spherical triangle \(\tilde{A}\tilde{B}\tilde{C}\) which its sides are great-circle-arcs. Full article
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