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Showing content from https://doi.org/10.1007/s00006-017-0798-7 below:

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

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.

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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)

2. Boubekeur, T., Alexa, M.: Phong tessellation. ACM Trans. Graphics (TOG) 27(5), 141 (2008)

   Google Scholar 

3. Crumeyrolle, A.: Orthogonal and symplectic clifford algebras. Springer (1991)

4. Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993)

   Article  ADS  MATH  MathSciNet  Google Scholar 

5. Plücker, J.: On a new geometry of space. Philosophical Transactions of the Royal Society of London, 155:725–791. The Royal Society (1865)

6. Gunn, C.: Guide to geometric algebra in practice, chapter on the homogeneous model of euclidean geometry, pp. 297–327. Spring, London (2011)

7. Gunn, C.: Geometric algebras for euclidean geometry. Adv. Appl. Clifford Algebras 27(1), 185–208 (2017)

   Article  MATH  MathSciNet  Google Scholar 

8. Dorst, L.: 3D oriented projective geometry through versors of \(R^{3,3}\). Adv. Appl. Clifford Algebras 26(4), 1137–1172 (2016)

   Article  MATH  MathSciNet  Google Scholar 

9. Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science. Morgan-Kaufmann (2007)

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)

11. Easter, R.B., Hitzer, E.: Conic and cyclidic sections in double conformal geometric algebra \(G_{8,2}\) Proceedings of SSI 2016 (2016)

12. Fontijne, D., Bouma, T., Dorst, L.: Geometric algebra implementation generator, Gaigen (2002)

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)

    Article  MATH  MathSciNet  Google Scholar 

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)

15. Klein, F.: Vorlesungen ueber hoehere Geometrie. Springer (1926)

16. Li, H., Huang, L., Shao, C., Dong, L.: Three-dimensional projective geometry with geometric algebra. arXiv preprint arXiv:1507.06634 (preprint)

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)

18. Parkin, S.T.: A model for quadric surfaces using geometric algebra. Unpublished, October (2012)

19. Perwass, C.: Geometric algebra with applications in engineering. Springer, Berlin (2009)

    MATH  Google Scholar 

20. Pottmann, H., Wallner, J.: Computational line geometry. Springer, Berlin (2001)

    MATH  Google Scholar 

21. Triggs, B.: Auto-calibration and the absolute quadric. Computer vision and pattern recognition, pp. 609–614 (1997)

22. Vince, J.A.: Geometric Algebra for computer graphics, 1st edn. Springer-Verlag TELOS, Santa Clara (2008)

    Book  MATH  Google Scholar 

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1. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, Jiangsu, China

   Juan Du

2. Department of Computer Science, Rice University, 6100 Main Street, Houston, 77005-1892, TX, USA

   Ron Goldman

3. Cheriton School of Computer Science, University of Waterloo, 200 University Ave. W, Waterloo, N2L 3G1, ON, Canada

   Stephen Mann

1. Juan Du
2. Ron Goldman
3. Stephen Mann

Correspondence to Stephen Mann.

Communicated by Leo Dorst

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

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• Received: 10 May 2016

• Accepted: 30 June 2017

• Published: 20 July 2017

• Issue Date: December 2017

• DOI: https://doi.org/10.1007/s00006-017-0798-7

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