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Search Results (4)

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Keywords = timelike axodes

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18 pages, 787 KiB  
Article
Kinematic Geometry of a Timelike Line Trajectory in Hyperbolic Locomotions
by Areej A. Almoneef and Rashad A. Abdel-Baky
Axioms 2023, 12(10), 915; https://doi.org/10.3390/axioms12100915 - 26 Sep 2023
Viewed by 1113
Abstract
This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical [...] Read more.
This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical locomotions. Subsequently, a timelike line congruence is defined and its spatial equivalence is thoroughly studied. The formulated assertions degenerate into a quadratic form, which facilitates a comprehensive understanding of the geometric features of the inflection line congruence. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
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15 pages, 639 KiB  
Article
A Study on a Spacelike Line Trajectory in Lorentzian Locomotions
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2023, 15(10), 1816; https://doi.org/10.3390/sym15101816 - 24 Sep 2023
Viewed by 968
Abstract
In this study, we establish a novel Lorentzian interpretation of the Euler–Savary (E−S) and Disteli (Dis) formulae. Subsequently, we proceed to establish a theoretical structure for a Lorentzian torsion line congruence which is the spatial [...] Read more.
In this study, we establish a novel Lorentzian interpretation of the Euler–Savary (E−S) and Disteli (Dis) formulae. Subsequently, we proceed to establish a theoretical structure for a Lorentzian torsion line congruence which is the spatial symmetry of the Lorentzian circling-point dual curve, in accordance with the principles of the kinematic theory of spherical locomotions. Further, a timelike (Tlike) torsion line congruence is defined and its spatial equivalence is examined. The findings contribute to an enhanced comprehension of the interplay between axodes and Lorentzian spatial movements, which has possible significance in various disciplines, such as the fields of robotics and mechanical engineering. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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13 pages, 479 KiB  
Article
One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode
by Areej A. Almoneef and Rashad A. Abdel-Baky
Mathematics 2023, 11(17), 3749; https://doi.org/10.3390/math11173749 - 31 Aug 2023
Viewed by 1064
Abstract
In this paper, based on the E. Study map, direct appearances were sophisticated for one-parameter hyperbolic dual spherical locomotions and invariants of the axodes. With the suggested technique, the Disteli formulae for the axodes were acquired and the correlations through kinematic geometry of [...] Read more.
In this paper, based on the E. Study map, direct appearances were sophisticated for one-parameter hyperbolic dual spherical locomotions and invariants of the axodes. With the suggested technique, the Disteli formulae for the axodes were acquired and the correlations through kinematic geometry of a timelike line trajectory were provided. Then, a ruled analogy of the curvature circle of a curve in planar locomotions was expanded into generic spatial locomotions. Lastly, we present new hyperbolic proofs for the Euler–Savary and Disteli formulae. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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16 pages, 774 KiB  
Article
Spacelike Lines with Special Trajectories and Invariant Axodes
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2023, 15(5), 1087; https://doi.org/10.3390/sym15051087 - 15 May 2023
Viewed by 1371
Abstract
The association between the instantaneous invariants of a one-parameter Lorentzian spatial movement and the spacelike lines with certain trajectories is considered in this study. To be more precise, we present a theoretical formulation of a Lorentzian inflection line congruence, which is the spatial [...] Read more.
The association between the instantaneous invariants of a one-parameter Lorentzian spatial movement and the spacelike lines with certain trajectories is considered in this study. To be more precise, we present a theoretical formulation of a Lorentzian inflection line congruence, which is the spatial symmetrical of the inflection circle of planar kinematics. Finally, we establish novel Lorentzian explanations for the Disteli and Euler–Savary formulae. Our results add to a better understanding of the interaction between axodes and Lorentzian spatial movements, with potential implications in fields such as robotics and mechanical engineering. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 2nd Edition)
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