“Extended Cross-Product” and Solution of a Linear System of Equations
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.
KeywordsLinear 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.
- 3.Dorst, L., Fontine, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco (2007)Google Scholar
- 4.Calvet, R.G.: Treatise of Plane Geometry through Geometric Algebra (2007)Google Scholar
- 8.Krumm, J.: Intersection of Two Planes, Microsoft Research, May 2000. http://research.microsoft.com/apps/pubs/default.aspx?id=68640
- 9.Johnson, M.: Proof by duality: or the discovery of “new” theorems. Math. Today 32(11), 171–174 (1996)Google Scholar
- 10.MacDonald, A.: Linear and Geometric Algebra. CreateSpace, Charleston (2011)Google Scholar
- 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.Skala, V.: Projective Geometry and Duality for Graphics, Games and Visualization - Course SIGGRAPH Asia 2012, Singapore (2012). ISBN:978-1-4503-1757-3Google Scholar
- 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