Abstract
Let \(L^{4}\) be a 4-dimensional Lorentzian space with the sign (−,+,+,+). The aim of this study is to investigate the other missing algebraic forms of the constraint manifolds of 2C and 3C spatial open chains in \(L^{4}\). For this purpose, firstly, we obtain the structure equations of a spatial open chain using the equations of open chains of the Lorentz plane and Lorentz sphere. After then, using these structure equations, we search the algebraic forms of the constraint manifolds of 2C and 3C spatial open chains in Lorentzian 3-space with respect to the causal characters of the first link and the axis of rotation of the joint.
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Aktaş, B., Durmaz, O., Gündoğan, H.: On constraint manifolds of Lorentz sphere: An. St. Univ. Ovidius Constanta. Ser. Mat. 28, 15–34 (2020)
Beggs, J.S.: Kinematics. ISBN: 0891163557: Taylor & Francis p.1 (1983)
Berenson, D., Srinivasa, S.S., Ferguson D., Kuffner, J.J.: Manipulation planning on constraint manifolds. In IEEE International Conference on Robotics and Automotion (2009)
Biewener, A.: Animal Locomotion. Oxford University Press (2003)
Blaschke, W.: Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie. Dover, New York (1945)
Bordalba, R., Ros, L., Porta, J.M.: Kinodynamic Planning on Constraint Manifolds. arXiv: 1705.07637v1. (2017)
Bottema, O., Roth, B.: Theoretical Kinematics. North-Hollanda Press, New York (1979)
Durmaz, O., Aktaş, B., Gündoğan, H.: The derivative and tangent operators of a motion in Lorentz space. Int. J. Geometr. Methods Mod. Phys. 19(4), 1–11 (2017)
Durmaz, O., Aktaş, B., Gündoğan, H.: Structure equations and constraint manifolds on Lorentz plane. Math. Methods Appl. Sci. 42(16), 5199–5214 (2019)
Durmaz, O., Aktaş, B., Keçilioğlu, O.: An overview to analyticity of dual functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 71(4), 1094–1119 (2022)
Fischer, I.S.: Dual-number methods in kinematics. CRC Press LLC, Statics and Dynamics (1999)
Gözütok, A.Ç., Ozkaldı Karakuş, S., Gündoğan, H.: Conics and quadrics in Lorentz space. Math. Appl. E-Notes. 6(1), 58–63 (2018)
Gündoğan, H., Keçilioğlu, O.: Lorentzian matrix multiplication and the motion on Lorentzian plane. Glass. Mat. 41, 329–334 (2006)
Herranz, F.J., Ortega, R., Santander, M.: Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry. J. Phys. A Math. Gen. 33(24), 4525 (2000)
Hibbeler, R.C.: Kinematics and kinetics of a particle. In Dynamics. Singapore Prentice Hall, Engineering Mechanics (2009)
Karakiliç, I., Gürsoy, A.E.: The dual exponential mapping on dual rotations. Sci. Res. Essays. 6(22), 4792–4797 (2011)
Keçilioğlu, O., Ozkaldı, S., Gündoğan, H.: Rotations and screw motion with Timelike vector in 3-Dimensional Lorentzian space. Adv. Appl. Clifford Algebra. 22, 1081–1091 (2012)
Knossow, D., Ronfard, R., Horaud, R.: Human motion tracking with a kinematic parametrization of extremal contours. Int. J. Comput. Vis. 79, 247–269 (2008)
Lopez, R.: Differential geometry of curves and surfaces in Lorentz–Minkowski space. Int. Electron. J. Geometry. 7(1), 44–107 (2014)
McCharthy, J.M.: An Introduction to Theoretical Kinematics. The MIT Press, Cambridge (1990)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press Inc, New York (1983)
Ozkaldı, S., Gündoğan, H.: Cayley formula, Euler parameters and rotations in in 3-Dimensional Lorentzian space. Adv. Appl. Clifford Algebra. 20, 367–377 (2010)
Ratcliffe, R.G.: Foundations of Hyperbolic Manifolds. Springer-Verlag, New York (1994)
Shabana, A.A.: Reference Kinematics. Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2003)
Teodorescu, P.P.: Kinematics, Mechanical Systems, Classical Models: Particle Mechanics. Springer, Dordrecht (2007)
Veldkamp, G.R.: On the use of dual numbers, vectors and matrices in instantenous spatial kinematics. Mech. Mach. Theory. 11(2), 141–156 (1976)
Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1904)
Wright, T.W.: Elements of Mechanics Including Kinematics, Kinetics and Statics. Chapter 1. New York, D Van Nostrand Company, London, E and FN Spon (1986)
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The author is deeply grateful to Prof. Halit Gündoğan and Dr. Buşra Aktaş for their highly valuable comments and suggestions.
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Durmaz, O. On Spatial Mechanisms in Lorentzian 3-Space. Mediterr. J. Math. 21, 69 (2024). https://doi.org/10.1007/s00009-024-02606-3
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DOI: https://doi.org/10.1007/s00009-024-02606-3