“Extended Cross-Product” and Solution of a Linear System of Equations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9786)


Many problems, not only in computer vision and visualization, lead to a system of linear equations Ax = 0 or Ax = b and fast and robust solution is required. A vast majority of computational problems in computer vision, visualization and computer graphics are three dimensional in principle. This paper presents equivalence of the cross–product operation and solution of a system of linear equations Ax = 0 or Ax = b using projective space representation and homogeneous coordinates. This leads to a conclusion that division operation for a solution of a system of linear equations is not required, if projective representation and homogeneous coordinates are used. An efficient solution on CPU and GPU based architectures is presented with an application to barycentric coordinates computation as well.


Linear system of equations Extended cross-product Projective space computation Geometric algebra Scientific computation 


  1. 1.
    Coxeter, H.S.M.: Introduction to Geometry. Wiley, New York (1961)MATHGoogle Scholar
  2. 2.
    Doran, Ch., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Dorst, L., Fontine, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco (2007)Google Scholar
  4. 4.
    Calvet, R.G.: Treatise of Plane Geometry through Geometric Algebra (2007)Google Scholar
  5. 5.
    Hartley, R., Zisserman, A.: MultiView Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  6. 6.
    Hildenbrand, D.: Foundations of Geometric Algebra Computing. Geometry and Computing. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Kanatani, K.: Understanding Geometric Algebra. CRC Press, Boca Raton (2015)CrossRefMATHGoogle Scholar
  8. 8.
    Krumm, J.: Intersection of Two Planes, Microsoft Research, May 2000. http://research.microsoft.com/apps/pubs/default.aspx?id=68640
  9. 9.
    Johnson, M.: Proof by duality: or the discovery of “new” theorems. Math. Today 32(11), 171–174 (1996)Google Scholar
  10. 10.
    MacDonald, A.: Linear and Geometric Algebra. CreateSpace, Charleston (2011)Google Scholar
  11. 11.
    Skala, V.: A new approach to line and line segment clipping in homogeneous coordinates. Vis. Comput. 21(11), 905–914 (2005)CrossRefGoogle Scholar
  12. 12.
    Skala, V.: Length, area and volume computation in homogeneous coordinates. Int. J. Image Graph. 6(4), 625–639 (2006)CrossRefGoogle Scholar
  13. 13.
    Skala, V.: Barycentric Coordinates Computation in Homogeneous Coordinates. Comput. Graph. 32(1), 120–127 (2008). ISSN 0097-8493MathSciNetCrossRefGoogle Scholar
  14. 14.
    Skala, V.: Projective geometry, duality and precision of computation in computer graphics, visualization and games. In: Tutorial Eurographics 2013, Girona (2013)Google Scholar
  15. 15.
    Skala, V.: Projective Geometry and Duality for Graphics, Games and Visualization - Course SIGGRAPH Asia 2012, Singapore (2012). ISBN:978-1-4503-1757-3Google Scholar
  16. 16.
    Skala, V.: Intersection computation in projective space using homogeneous coordinates. Int. J. Image Graph. 8(4), 615–628 (2008)CrossRefGoogle Scholar
  17. 17.
    Skala, V.: Modified gaussian elimination without division operations. In: ICNAAM 2013, AIP Conference Proceedings Rhodos, Greece, no. 1558, pp. 1936–1939. AIP Publishing (2013)Google Scholar
  18. 18.
    Vince, J.: Geometric Algebra for Computer Science. Springer, London (2008)MATHGoogle Scholar
  19. 19.
    Wildberger, N.J.: Divine Proportions: Rational Trigonometry to Universal Geometry. Wild Egg Pty, Sydney (2005)MATHGoogle Scholar
  20. 20.
    Yamaguchi, F.: Computer Aided Geometric Design: A totally Four Dimensional Approach. Springer, Tokyo (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Applied SciencesUniversity of West BohemiaPlzenCzech Republic

Personalised recommendations