Skip to main content
Log in

Global Invariants of Paths and Curves in the Two-Dimensional Real Vector Space for Linear Similarity Groups Generated by Dual Numbers

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

Let \(\mathbb {D}\) be the algebra of dual numbers and \(G=M^{+}(\mathbb {D^{*}}), M(\mathbb {D^{*}})\) be linear similarity groups generated by the algebra \(\mathbb {D}\) in two-dimensional real vector space \(R^{2}\). The present paper is devoted to solutions of problems of global G-equivalence of paths and curves in \(R^{2}\) for groups \(G=M^{+}(\mathbb {D^{*}}), M(\mathbb {D^{*}})\). Complete systems of global G-invariants of a path and a curve in \(R^{2}\) are obtained. Existence and uniqueness theorems are given. Evident forms of a path and a curve with the given complete system of G-invariants are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrawal, S.K.: Multibody dynamics: a formulation using Kanes method and dual vectors. ASME J. Mech. Des. 115, 833–838 (1993)

    Article  Google Scholar 

  2. Angeles, J.: The application of dual algebra to kinematic analysis. In: Angeles, J., Zakhariev, E. (eds.) Computational Methods in Mechanical Systems, p. 332. Springer, New York (1991)

    Google Scholar 

  3. Aripov, R.G., Khadjiev, D.: The complete system of global differential and integral invariants of a curve in Euclidean geometry. Russian Math. (Iz VUZ) 51(7), 1–14 (2007)

    Article  MathSciNet  Google Scholar 

  4. Bagci, C.: Dynamic force and torque analysis for mechanisms using dual vectors and \(3\times 3\) screw matrix. ASME J. Eng. Ind. 94, 738–745 (1972)

    Article  Google Scholar 

  5. Ball, R.S.: Theory of Screws. Cambridge University Press, Cambridge (1900)

    MATH  Google Scholar 

  6. Cheng, H.H.: Programming with dual numbers and its applications in mechanisms design. Eng. Comput. 10(4), 212–229 (1994)

    Article  Google Scholar 

  7. Clifford, W.K.: Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. 4(64), 381–395 (1873)

    MathSciNet  MATH  Google Scholar 

  8. Daher, M.: Dual Numbers and Invariant Theory of the Euclidean group with applications to robotics. A thesis, Doctor of Philosophy in Mathematics, Victoria University of Wellington, (2013)

  9. Denavit, J.: Displacement analysis of mechanisms on \(2\times 2\) matrices of dual numbers. VDI Ber. 29, 81–88 (1958)

    Google Scholar 

  10. Dimentberg, F. M.: The Screw Calculus and its Applications in Mechanics. (Izdat. ’Nauka”, Moscow, USSR, 1965) English translation: AD 680993, Clearinghouse for Federal and Scientific Technical Information

  11. Encheva, R.P., Georgiev, G.H.: Similar Frenet Curves. Result. Math. 55, 359–372 (2009)

    Article  MathSciNet  Google Scholar 

  12. Fischer, I.S.: Dual-Number Methods in Kinematics, Statics and Dynamics. CRC Press, Boca Raton London New York Washington D.C. (1999)

  13. Gu, Y.-L., Luh, J.Y.S.: Dual-number transformations and its applications to robotics. IEEE J. Robot. Autom. RA-3, 615–623 (1987)

  14. Keler, M.L.: Kinematics and statics including friction in single-loop mechanisms by screw calculus and dual vectors. ASME J. Eng. Ind. 95(2), 471–480 (1973)

    Article  Google Scholar 

  15. Khadjiev, D., Ören, I., Pekşen, Ö.: Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry. Turkish J. Math. 37, 80–94 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Khadjiev, D., Pekşen, Ö.: The complete system of global differential and integral invariants of equiaffne curves. Diff. Geom. And Appl. 20, 168–175 (2004)

    Article  Google Scholar 

  17. Kőse, O.: Kinematic differential geometry of a rigid body in spatial motion using dual vector calculus: Part-I. Appl. Math. Comput. 183, 17–29 (2006)

    MathSciNet  MATH  Google Scholar 

  18. McCarthy, J. M.: On the scalar and dual formulations of the curvature theory of line trajectories. ASME J. Mech. Transm. Autom. Des. 109/101, 97–106 (1987)

  19. Moon, Y.-M., Kota, S.: Automated synthesis of mechanisms using dual-vector algebra. Mech. Mach. Theory 37, 143–166 (2002)

    Article  Google Scholar 

  20. Özdemir, M., Şimsek, H.: Similar and self-similar curves in Minkowski n-space. Bull. Korean Math. Soc. 52(6), 20712093 (2015)

    Article  MathSciNet  Google Scholar 

  21. Pennestri, E., Stefanelli, R.: Linear algebra and numerical algorithms using dual numbers. Multibody Syst. Dyn. 18, 323–344 (2007)

    Article  MathSciNet  Google Scholar 

  22. Pennock, G.R., Yang, A.T.: Application of dual-number matrices to the inverse kinematics problem of robot manipulators. ASME J. Mech. Transm. Autom. Des. 107, 201–208 (1985)

    Article  Google Scholar 

  23. Rooney, J.: On the principle of transference. In: Proc. of the IV IFToMM Congress, New Castle Upon Tyne, UK, pp. 1089–1094 (1975)

  24. Spall, R.E., Yu, W.: Imbedded dual-number automatic differentiation for CFD sensitivity analysis. In: Proceedings of the Fluids Engineering Division Summer Conference, San Juan, Puerto Rico, July, 08–12 (2012)

  25. Study, E.: Geommetry der Dynamen. Leipzig (1901)

  26. Turgut, M.: On invariants of time-like dual curves. Hacettepe J. Math. Stat. 37(2), 129–133 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Veldkamp, G.R.: On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics. Mech. Mach. Theory 11(2), 141–156 (1976)

    Article  Google Scholar 

  28. Yaglom, I.M.: Complex numbers in geometry. Academic Press, New York and London (1964)

    MATH  Google Scholar 

  29. Yang, A.T., Freudenstein, F.: Application of dual number quaternions algebra to the analysis of spatial mechanisms. ASME J. Eng. Ind. 86, 300–308 (1964)

    MathSciNet  MATH  Google Scholar 

  30. Yang, A.T.: Analysis of an offset unsymmetric gyroscope with oblique rotor using \((3\times 3)-\) matrices with dual-number elements. ASME J. Eng. Ind. 91(3), 535–542 (1969)

    Article  Google Scholar 

  31. Yu, W., Blair, M.: DNAD, a Simple Tool for Automatic Differentiation of Fortran Codes Using Dual Numbers. Utah State University, Mechanical and Aerospace Engineering Faculty Publications. Paper 30 (2013)

  32. Yücesan, A., Ayyildiz, N., Çöken, A.C.: On rectifying dual space curves. Rev. Mat. Complut. 20(2), 497–506 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is very grateful to the reviewer(s) for helpful comments and valuable suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yasemin Sağiroğlu.

Additional information

Communicated by Hongbo Li

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sağiroğlu, Y. Global Invariants of Paths and Curves in the Two-Dimensional Real Vector Space for Linear Similarity Groups Generated by Dual Numbers. Adv. Appl. Clifford Algebras 29, 10 (2019). https://doi.org/10.1007/s00006-018-0929-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-018-0929-9

Keywords

Mathematics Subject Classification

Navigation