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

  • Vaclav Skala
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 



The author would like to thank to colleagues at the University of West Bohemia in Plzen for fruitful discussions and to anonymous reviewers for their comments and hints which helped to improve the manuscript significantly. Special thanks belong also to SIGGRAPH and Eurographics tutorials attendee for their constructive questions, which stimulated this work.

This research was supported by the MSMT CZ - project No. LH12181.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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