Skip to main content
SpringerLink
Log in

A Simple Derivation of the Gauss-Bonet Theorem

  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

Abstract

A simple derivation of the Gauss-Bonet theorem is presented based on the representation of spherical polygons by Euler angles and Rodrigues transposition theorem. This leads to a derivation of the theorem which avoids completely the explicit evaluation of rotation matrices.

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. MUKHERJEE, R. and PUKRUSHPAN, J. T. “Class of Rotations Induced by Spherical Polygons,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 23, No. 4, 2000, pp. 746–749.

    Article  Google Scholar 

  2. LEVI, M. “Geometric Phases in the Motion of Rigid Bodies,” Archive for Rational Mechanics, Vol. 122, No. 3, 1999, pp. 213–219.

    Article  MathSciNet  Google Scholar 

  3. SHUSTER, M. D. “A Survey of Attitude Representations,” The Journal of the Astronautical Sciences, Vol. 41, No. 4, 1993, pp. 439–517.

    MathSciNet  Google Scholar 

  4. JUNKINS, J. L. and SHUSTER, M. D. “The Geometry of the Euler Angles,” The Journal of the Astronautical Sciences, Vol. 41, No. 4, 1993, pp. 531–543.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Michigan State University, East Lansing, MI, 48824-1226, USA

    Ranjan Mukherjee (Associate Professor)

  2. Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, MD, 20874, USA

    Malcolm D. Shuster (Director of Research)

Authors
  1. Ranjan Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Malcolm D. Shuster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ranjan Mukherjee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, R., Shuster, M.D. A Simple Derivation of the Gauss-Bonet Theorem. J of Astronaut Sci 53, 185–191 (2005). https://doi.org/10.1007/BF03546349

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03546349

Search

Navigation