Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 661)


This contribution describes a new approach for solving linear system of algebraic equations and differential equations using Laplace transform by the extended-cross product. It will be shown that a solution of a linear system of equations Ax = 0 or Ax = b is equivalent to the extended cross-product if the projective extension of the Euclidean system and the principle of duality are used. Using the Laplace transform differential equations are transformed to a system of linear algebraic equations, which can be solved using the extended cross-product (outer product). The presented approach enables to avoid division operation and extents numerical precision as well. It also offers applications of matrix-vector and vector-vector operations in symbolic manipulation, which can leads to new algorithms and/or new formula. The proposed approach can be applied also for stability evaluation of dynamical systems. In the case of numerical computation, it supports vector operation and SSE instructions or GPU can be used efficiently.


Linear system of equations Linear system of differential equations Laplace transform Extended cross product Outer product Homogeneous coordinates Duality Geometrical algebra Dynamic systems Stability GPGPU computation SSE instructions 


  1. 1.
    Coxeter, H.S.M.: Introduction to Geometry. Wiley, New York (1961)MATHGoogle Scholar
  2. 2.
    Dorst, L., Fontine, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco (2007)Google Scholar
  3. 3.
    Duffy, D.G.: Transform Methods for Solving Partial Differential Equations. CRC Press, Boca Raton (1994)MATHGoogle Scholar
  4. 4.
    Franklin, G., Powell, D., Emami-Naeini, A.: Feedback Control of Dynamic Systems. Prentice-Hall, Englewood Cliffs (2002)MATHGoogle Scholar
  5. 5.
    Calvet, R.G.: Treatise of Plane Geometry Through Geometric Algebra (2007)Google Scholar
  6. 6.
    Hildenbrand, D.: Foundations of Geometric Algebra Computing. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Johnson, M.: Proof by duality: or the discovery of “new” theorems. Math. Today 32(11), 171–174 (1996)Google Scholar
  8. 8.
    Kanatani, K.: Undestanding Geometric Algebra. CRC Press, Boca Raton (2015)Google Scholar
  9. 9.
    MacDonald, A.: Linear and Geometric Algebra. CreateSpace, Charleston (2011)Google Scholar
  10. 10.
    Skala, V.: A new approach to line and line segment clipping in homogeneous coordinates. Vis. Comput. 21(11), 905–914 (2005). SpringerCrossRefGoogle Scholar
  11. 11.
    Skala, V.: Length, area and volume computation in homogeneous coordinates. Int. J. Image Graph. 6(4), 625–639 (2006)CrossRefGoogle Scholar
  12. 12.
    Skala, V.: Barycentric coordinates computation in homogeneous coordinates. Comput. Graph. 32(1), 120–127 (2008)CrossRefGoogle Scholar
  13. 13.
    Skala, V.: Intersection computation in projective space using homogeneous coordinates. Int. J. Image Graph. 8(4), 615–628 (2008)CrossRefGoogle 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. In: Course SIGGRAPH Asia 2012, Singapore (2012). ISBN 978-1-4503-1757-3Google Scholar
  16. 16.
    Skala, V.: Modified Gaussian elimination without division operations. In: ICNAAM 2013, Rhodos, Greece, AIP Conference Proceedings, No. 1558, pp. 1936–1939. AIP Publishing (2013) Google Scholar
  17. 17.
    Skala, V.: “Extended cross-product” and solution of a linear system of equations. In: Gervasi, O., et al. (eds.) Computational Science and Its Applications – ICCSA 2016. LNCS, vol. 9786, pp. 18–35. Springer, Cham (2016)CrossRefGoogle Scholar
  18. 18.
    Skala, V.: Plücker coordinates and extended cross product for robust and fast intersection computation. In: CGI 2016 & CGI 2016 Proceedings, pp. 57–60. ACM, Greece (2016)Google Scholar
  19. 19.
    Vince, J.: Geometric Algebra for Computer Science. Springer (2008)Google Scholar
  20. 20.
    Yamaguchi, F.: Computer Aided Geometric Design: A Totally Four Dimensional Approach. Springer, Tokyo (2002)CrossRefGoogle Scholar
  21. 21.
  22. 22.
    Solving Systems of ODEs via the Laplace Transform.
  23. 23.
  24. 24.
  25. 25.
    Rosenbrock, H.H.: State-Space and Multivariable Theory. Nelson, London (1970)MATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Faculty of Applied SciencesUniversity of West BohemiaPlzenCzech Republic

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