Advances in Applied Clifford Algebras

, Volume 27, Issue 4, pp 3039–3062 | Cite as

Modeling 3D Geometry in the Clifford Algebra R(4, 4)

  • Juan Du
  • Ron Goldman
  • Stephen Mann


We flesh out the affine geometry of \({{\mathbb {R}}^3}\) represented inside the Clifford algebra \({\mathbb {R}}(4,4)\). We show how lines and planes as well as conic sections and quadric surfaces are represented in this model. We also investigate duality between different representations of points, lines, and planes, and we show how to represent intersections between these geometric elements. Formulas for lengths, areas, and volumes are also provided.


Mother algebra Affine geometry Computer graphics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bayro-Corrochano, E., Lasenby, J.: Algebraic frames for the perception-action cycle, chapter A unified language for computer vision and robotics, pp. 219–234. Springer, Berlin (1997)Google Scholar
  2. 2.
    Boubekeur, T., Alexa, M.: Phong tessellation. ACM Trans. Graphics (TOG) 27(5), 141 (2008)Google Scholar
  3. 3.
    Crumeyrolle, A.: Orthogonal and symplectic clifford algebras. Springer (1991)Google Scholar
  4. 4.
    Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Plücker, J.: On a new geometry of space. Philosophical Transactions of the Royal Society of London, 155:725–791. The Royal Society (1865)Google Scholar
  6. 6.
    Gunn, C.: Guide to geometric algebra in practice, chapter on the homogeneous model of euclidean geometry, pp. 297–327. Spring, London (2011)Google Scholar
  7. 7.
    Gunn, C.: Geometric algebras for euclidean geometry. Adv. Appl. Clifford Algebras 27(1), 185–208 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dorst, L.: 3D oriented projective geometry through versors of \(R^{3,3}\). Adv. Appl. Clifford Algebras 26(4), 1137–1172 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science. Morgan-Kaufmann (2007)Google Scholar
  10. 10.
    Easter, R.B., Hitzer, E.: Double conformal geometric algebra for quadrics and darboux cyclides. Proceedings of Computer Graphics International Conference, Heraklion, Greece, CFI ’16, 93–96 (2016)Google Scholar
  11. 11.
    Easter, R.B., Hitzer, E.: Conic and cyclidic sections in double conformal geometric algebra \(G_{8,2}\) Proceedings of SSI 2016 (2016)Google Scholar
  12. 12.
    Fontijne, D., Bouma, T., Dorst, L.: Geometric algebra implementation generator, Gaigen (2002)Google Scholar
  13. 13.
    Goldman, R., Mann, S.: R(4,4) as a computational framework for 3-dimensional computer graphics. Adv. Appl. Clifford Algebras 25(1), 113–149 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hestenes, D., Li, H., Rockwood, A.: Geometric computing with Clifford algebras, chapter New algebraic tools for classical geometry, pp. 3–26. Springer, Berlin Heidelberg (2011)Google Scholar
  15. 15.
    Klein, F.: Vorlesungen ueber hoehere Geometrie. Springer (1926)Google Scholar
  16. 16.
    Li, H., Huang, L., Shao, C., Dong, L.: Three-dimensional projective geometry with geometric algebra. arXiv preprint arXiv:1507.06634 (preprint)
  17. 17.
    Li, H., Zhang, L.: Guide to geometric algebra in practice, chapter Line Geometry in Terms of the Null Geometric Algebra over \({R^{3,3}}\) and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms, pp. 253–272. Spring, London (2011)Google Scholar
  18. 18.
    Parkin, S.T.: A model for quadric surfaces using geometric algebra. Unpublished, October (2012)Google Scholar
  19. 19.
    Perwass, C.: Geometric algebra with applications in engineering. Springer, Berlin (2009)zbMATHGoogle Scholar
  20. 20.
    Pottmann, H., Wallner, J.: Computational line geometry. Springer, Berlin (2001)zbMATHGoogle Scholar
  21. 21.
    Triggs, B.: Auto-calibration and the absolute quadric. Computer vision and pattern recognition, pp. 609–614 (1997)Google Scholar
  22. 22.
    Vince, J.A.: Geometric Algebra for computer graphics, 1st edn. Springer-Verlag TELOS, Santa Clara (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.College of Mechanical and Electrical EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.Department of Computer ScienceRice UniversityHoustonUSA
  3. 3.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations