Skip to main content
SpringerLink
Log in

Using Dual Quaternion to Study Translational Surfaces

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms.

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

Access this article

Log in via an institution

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. Altman, S.L.: Rotations, Quaternions, and Double Groups. Dover Publications, Mineola (1986)

    Google Scholar 

  2. Clifford, W.K.: Preliminary sketch of bi-quaternions. Proc. Lond. Math. Soc. 1–4, 381–395 (1873)

    MATH  Google Scholar 

  3. Dai, J.S.: An historical review of the theoretical development of rigid body displacements from rodrigues parameters to the finite twist. Mech. Mach. Theory 41, 41–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Daniilidis, K.: Hand-eye calibration using dual quaternions. Int. J. Robot. Res. 18, 286–298 (1999)

    Article  Google Scholar 

  5. Fischer, I.: Dual-Number Method in Kinematics, Statics and Dynamics. CRC Press, Boca Raton (1998)

    Google Scholar 

  6. Funda, J., Paul, R.P.: A computational analysis of screw transformations in robotics. IEEE Trans. Robot. Autom. 6, 348–356 (1990)

    Article  Google Scholar 

  7. Goldman, R.: Rethinking Quaternions: Theory and Computation. In: Barsky, Brian A. (ed.) Synthesis Lectures on Computer Graphics and Animation, vol. 13. Morgan & Claypool Publishers, San Rafael (2010)

    Google Scholar 

  8. Hamilton, W.: Elements of Quaternions. Cambridge University Press, Cambridge (1866)

    Google Scholar 

  9. Jüttler, B.: Visualization of moving objects using dual quaternion curves. Comput. Graph. 18, 315–326 (1994)

    Article  Google Scholar 

  10. McCarthy, M.: Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990)

    Google Scholar 

  11. Pérez-Díaz, S., Shen, L.: Parametrization of translational surfaces. In: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, pp. 128–129 (2014)

  12. Schroeder, W.J., Lorensen, W.E., Linthicum, S.: Implicit modeling of swept surfaces and volumes. In: Proceedings of the Conference on Visualization’94, pp. 40–45. IEEE Computer Society Press (1994)

  13. Zatsiorsky, V.M.: Kinematics of Human Motion. Human Kinetics, Champaign, IL (1998)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO, 63701, USA

    Haohao Wang

  2. Department of Computer Science, Rice University, Houston, TX, 77251, USA

    Ron Goldman

Authors
  1. Haohao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ron Goldman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haohao Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Goldman, R. Using Dual Quaternion to Study Translational Surfaces. Math.Comput.Sci. 12, 69–75 (2018). https://doi.org/10.1007/s11786-018-0330-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-018-0330-z

Keywords

Mathematics Subject Classification

Search

Navigation