One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode
Abstract
:1. Introduction
2. Basic Concepts
- (i)
- If and are two spacelike dual vectors, then
- •
- If they specify a spacelike dual plane, there is a single dual number ; , and such that . This number is the spacelike dual angle through and .
- •
- If they specify a timelike dual plane, there is a single dual number such that , where or via or , respectively. This number is the central dual angle through and .
- (ii)
- If and are two timelike dual vectors, then there is a single dual number such that , where or via and have different time-direction or the same time-direction, respectively. This dual number is the Lorentzian timelike dual angle through and .
- (iii)
- If is spacelike dual, and is timelike dual, then there is a single dual number 0 such that , where or via or . This number is the Lorentzian timelike dual angle through and .
3. One-Parameter Hyperbolic Dual Spherical Locomotions
3.1. Spatial Kinematics and Invariants of the Axodes
3.2. Euler–Savary Formula for the Timelike Axodes
Velocity and Acceleration for a Timelike Line Trajectory
3.3. Euler–Savary and Disteli Formulae for ()
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Almoneef, A.A.; Abdel-Baky, R.A. One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode. Mathematics 2023, 11, 3749. https://doi.org/10.3390/math11173749
Almoneef AA, Abdel-Baky RA. One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode. Mathematics. 2023; 11(17):3749. https://doi.org/10.3390/math11173749
Chicago/Turabian StyleAlmoneef, Areej A., and Rashad A. Abdel-Baky. 2023. "One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode" Mathematics 11, no. 17: 3749. https://doi.org/10.3390/math11173749