Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra

Abstract

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. This leads to unification of PGA and CGA primitives which is important especially for software implementation and symbolic calculations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science: an object-oriented approach to geometry. Morgan Kaufmann Publishers Inc., Burlington (2007)

    Google Scholar 

  2. 2.

    Du, J., Goldman, R., Mann, S.: Modeling 3D Geometry in the Clifford Algebra R(4, 4). Adv. Appl. Clifford Algebras 27, 3039–3062 (2017). https://doi.org/10.1007/s00006-017-0798-7

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Gunn C.: On the Homogeneous model of euclidean geometry. In: Dorst L., Lasenby J. (eds) Guide to geometric algebra in practice. Springer, London (2011). https://doi.org/10.1007/978-0-85729-811-9_15

  4. 4.

    Gunn, C.G.: Doing euclidean plane geometry using projective geometric algebra. Adv. Appl. Clifford Algebras 27, 1203 (2017). https://doi.org/10.1007/s00006-016-0731-5

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Gunn, C.G.: Geometric algebras for Euclidean geometry. Adv. Appl. Clifford Algebras 27, 185 (2017). https://doi.org/10.1007/s00006-016-0647-0

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Gunn, C.G., De Keninck, S.: 3D PGA Cheat Sheet, An extensive reference with 3D PGA formulas, online: bivector.net

  7. 7.

    De Keninck, S.: ganja.js, https://zenodo.org/record/3635774 (2020)

  8. 8.

    Hildenbrand, D.: Foundations of geometric algebra computing. Springer Science & Business Media (2013)

  9. 9.

    Hildenbrand, D.: Introduction to geometric algebra computing. Chapman and Hall/CRC, Boca Raton (2018)

    Google Scholar 

  10. 10.

    Hrdina, J., Návrat, A., Vašík, P.: Control of 3-link robotic snake based on conformal geometric algebra. Adv. Appl. Clifford Algebr. 26(3), 1069–1080 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hrdina, J., Návrat, A., Vašík, P., Matoušek, R.: CGA-based robotic snake control. Adv. Appl. Clifford Algebr. 27(1), 621–632 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lasenby, A.: Rigid body dynamics in a constant curvature space and the ‘1d-up’ approach to conformal geometric algebra. In: Dorst, L., Lasenby, J. (eds.) Guide to geometric algebra in practice. Springer, London (2011)

    Google Scholar 

  13. 13.

    Lounesto, P.: Clifford algebra and spinors, 2nd edn. CUP, Cambridge (2006)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Aleš Návrat.

Additional information

Publisher's Note

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

The first three authors were supported by a Grant No.: FSI-S-20-6187.

Communicated by Dietmar Hildenbrand.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hrdina, J., Návrat, A., Vašík, P. et al. Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra. Adv. Appl. Clifford Algebras 31, 18 (2021). https://doi.org/10.1007/s00006-021-01118-7

Download citation

Keywords

  • Conformal geometric algebra
  • Projective geometric algebra
  • Euclidean geometry

Mathematics Subject Classification

  • Primary 15A66
  • Secondary 51N25