Kinematic-geometry of lines with special trajectories in spatial kinematics

Nadia Alluhaibi; Rashad A. Abdel-Baky; Nadia Alluhaibi; Rashad A. Abdel-Baky

doi:10.3934/math.2023552

AIMS Mathematics

2023, Volume 8, Issue 5: 10887-10904. doi: 10.3934/math.2023552

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Kinematic-geometry of lines with special trajectories in spatial kinematics

Nadia Alluhaibi ¹,
Rashad A. Abdel-Baky ^{2
,
,}

1.
Department of Mathematics, Science and Arts College, King Abdulaziz University, Rabigh, Kingdom of Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt

Received: 09 December 2022 Revised: 15 February 2023 Accepted: 27 February 2023 Published: 07 March 2023
MSC : 53A04, 53A05, 53A17

This paper derives the expressions for kinematic-geometry of lines with special trajectories in spatial kinematics by means of the E. Study map. A particular assurance goes to the 2nd order movement characteristics for extracting a new proof of the Disteli formulae of the axodes. Meanwhile, a new height dual function is defined and utilized to investigate the geometrical properties of the Disteli-axis. Consequently, as an implementation, the spatial equivalent of the cubic of stationary curvature is established and researched. Finally, Disteli formulae of a line-trajectory are extracted and inspected in detail.
- E. Study's map,
- axodes,
- line congruence,
- Disteli's formulae,
- height dual functions
Citation: Nadia Alluhaibi, Rashad A. Abdel-Baky. Kinematic-geometry of lines with special trajectories in spatial kinematics[J]. AIMS Mathematics, 2023, 8(5): 10887-10904. doi: 10.3934/math.2023552

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Abstract

This paper derives the expressions for kinematic-geometry of lines with special trajectories in spatial kinematics by means of the E. Study map. A particular assurance goes to the 2nd order movement characteristics for extracting a new proof of the Disteli formulae of the axodes. Meanwhile, a new height dual function is defined and utilized to investigate the geometrical properties of the Disteli-axis. Consequently, as an implementation, the spatial equivalent of the cubic of stationary curvature is established and researched. Finally, Disteli formulae of a line-trajectory are extracted and inspected in detail.

References

[1]	O. Bottema, B. Roth, Theoretical kinematics, New York: North-Holland Press, 1979.
[2]	A. Karger, J. Novak, Space kinematics and Lie groups, New York: Gordon and Breach Science Publishers, 1985.
[3]	J. Schaaf, Curvature theory of line trajectories in spatial kinematics, Ph. D Thesis, University of California, 1988.
[4]	H. Stachel, Instantaneous Spatial kinematics and the invariants of the axodes, Proceedings of A Symposion Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball Upon the 100th Anniversary of A Treatise on the Theory of Screws, 2000, 1–14.
[5]	H. Pottman, J. Wallner, Computational line geometry, Berlin: Springer-Verlag, 2001. http://dx.doi.org/10.1007/978-3-642-04018-4
[6]	R. Garnier, Cours de cinématique, Tome II: Roulement et vibration-la formule de Savary et son extension a l'espace, Paris: Gauthier-Villars, 1941.
[7]	J. Phillips, On the theorem of three axes in the spatial motion of three bodies, Aust. J. Appl. Sci., 15 (1964), 267.
[8]	M. Skreiner, A study of the geometry and the kinematics of Instantaneous spatial motion, J. Mech., 1 (1966), 115–143. http://dx.doi.org/10.1016/0022-2569(66)90017-6 doi: 10.1016/0022-2569(66)90017-6
[9]	B. Dizioglu, Einfacbe herleitung der Euler-Savaryschen konstruktion der riiumlichen bewegung, Mech. Mach. Theory, 9 (1974), 247–254. http://dx.doi.org/10.1016/0094-114X(74)90042-1 doi: 10.1016/0094-114X(74)90042-1
[10]	R. Abdel-Baky, F. Al-Solamy, A new geometrical approach to one-parameter spatial motion, J. Eng. Math., 60 (2008), 149–172. http://dx.doi.org/10.1007/s10665-007-9139-5 doi: 10.1007/s10665-007-9139-5
[11]	G. Figlioini, H. Stachel, J. Angeles, The computational fundamentals of spatial cycloidal gearing, In: Computational kinematics, Berlin: Springer, 2009,375–384. http://dx.doi.org/10.1007/978-3-642-01947-0_46
[12]	R. Abdel-Baky, R. Al-Ghefari, On the one-parameter spherical dual motions, Comput. Aided Geom. D., 28 (2011), 23–37. http://dx.doi.org/10.1016/j.cagd.2010.09.007 doi: 10.1016/j.cagd.2010.09.007
[13]	R. Al-Ghefari, R. Abdel-Baky, Kinematic geometry of a line trajectory in spatial motion, J. Mech. Sci. Technol., 29 (2015), 3597–3608. http://dx.doi.org/10.1007/s12206-015-0803-9 doi: 10.1007/s12206-015-0803-9
[14]	R. Abdel-Baky, On the curvature theory of a line trajectory in spatial kinematics, Commun. Korean Math. S., 34 (2019), 333–349. http://dx.doi.org/10.4134/CKMS.c180087 doi: 10.4134/CKMS.c180087
[15]	M. Aslan, G. Sekerci, Dual curves associated with the Bonnet ruled surfaces, Int. J. Geom. Methods M., 17 (2020), 2050204. http://dx.doi.org/10.1142/S0219887820502047 doi: 10.1142/S0219887820502047
[16]	N. Alluhaibi, Ruled surfaces with constant Disteli-axis, AIMS Mathematics, 5 (2020), 7678–7694. http://dx.doi.org/10.3934/math.2020491 doi: 10.3934/math.2020491
[17]	R. Abdel-Baky, F. Tas, W-Line congruences, Commun. Fac. Sci. Univ., 69 (2020), 450–460. http://dx.doi.org/10.31801/cfsuasmas.550369
[18]	R. Abdel-Baky, M. Naghi, A study on a line congruence as surface in the space of lines, AIMS Mathematics, 6 (2021), 11109–11123. http://dx.doi.org/10.3934/math.2021645 doi: 10.3934/math.2021645
[19]	J. Bruce, P. Giblin, Curves and singularities, 2 Eds., Cambridge: Cambridge University Press, 1992. http://dx.doi.org/10.1017/CBO9781139172615

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