Computer Graphics Techniques in Military Applications

  • Dimitrios Christou
  • Antonios Danelakis
  • Marilena Mitrouli
  • Dimitrios Triantafyllou
Part of the Springer Optimization and Its Applications book series (SOIA, volume 71)


The determination of intersection points of plane curves is a problem of Computer Graphics with many applications in Applied Mathematics, Numerical Analysis and many other scientific fields. More precisely, in military applications, the trajectories of two flying objects such as missiles, aircrafts etc, can be interpreted by two plane curves. Our scope is to find the intersection points of the given curves. The number of floating point operations (flops) of many classical methods is not satisfactory, since they demand over O(n 4) operations. Conversely, many algorithms that are fast enough, have serious problems with their numerical stability. The main objective here is to develop fast and stable algorithms computing the intersection points of plane curves. The error analysis and the computation of complexity of all the proposed methods are analysed and demonstrated through various examples.


Intersection points Plane curves Symbolic-numeric computations Modified Sylvester matrix Singular value decomposition 



The fourth author (D.T.) acknowledges financial support from State Scholarships Foundation (IKY).


  1. 1.
    Barnett, S.: greatest common divisor from generalized sylvester resultant matrices. Linear and Multilinear Algebra. 8, 271–279 (1980)MATHCrossRefGoogle Scholar
  2. 2.
    Conte, S.D., Carl de Boor: Elementary Numerical Analysis: An Algorithmic Approach, 3rd edn. pp. 74–80. McGraw-Hill Book Company, New York (1980)Google Scholar
  3. 3.
    Datta, B. N.: Numerical Linear Algebra and Applications, 2nd edn. SIAM, USA (2010)MATHCrossRefGoogle Scholar
  4. 4.
    Marco, A., Martinez, J-J.: A new source of structured singular value decomposition problems. Electronic Trans. Numer. Analy. 18, 188–197 (2004)Google Scholar
  5. 5.
    Sendra, J.R., Winkler, F.: Tracing index of rational curve parameterizations. Comput. Aided Geomet. Des. 18, 771–795 (2001)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Dimitrios Christou
    • 1
  • Antonios Danelakis
    • 2
  • Marilena Mitrouli
    • 3
  • Dimitrios Triantafyllou
    • 4
  1. 1.School of Engineering and Mathematical Sciences, Systems and Control CentreCity UniversityLondonUK
  2. 2.Department of Informatics and TelecommunicationsUniversity of AthensAthensGreece
  3. 3.Department of MathematicsUniversity of AthensAthensGreece
  4. 4.Department of Military Sciences, Section of Mathematics and Engineering SciencesHellenic Army AcademyVariGreece

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