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Minkowski Metrics: Its Interaction and Complementarity with Euclidean Metrics

  • Jean-Marie Becker
  • Michel Goeb
Chapter

Abstract

A certain set of objects has a Minkowski space structure when these objects can be represented by coordinates (x 0, x 1, , x n ) with \(({x}_{1}^{2} + \cdots + {x}_{n}^{2}) - {x}_{0}^{2} \leq 0\). This chapter considers three situations where this structure and its associated tools bring a new light. It first introduces a well-known such space, having its roots in Minkowski’s work, the space–time representation. Then, it considers a 3D representation space, S 2 +, for 2 ×2 symmetrical positive definite matrices, the determinant of these matrices providing this space with a Minkowski structure. An application to a representation of discrete curves is detailed. The last example is another 3D representation space, denoted Ω 2, for plane circles in the Euclidean space 2. The natural Minkowski quadratic form for a circle in Ω 2 is the square of its radius, for a convenient set of coordinates. An application is given to a new measure of circularity for a finite set of points.

References

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    Anderson I.M. and Bezdek J.C. (1984) “Curvature and tangential deflection of discrete arcs: a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data”. IEEE Trans. PAMI 6:27–40CrossRefMATHGoogle Scholar
  2. 2.
    Becker J.M. (1991) “Contribution à l’analyse d’images par marches quadrantales”, Ph. D. Thesis, University of Saint-Etienne, FranceGoogle Scholar
  3. 3.
    Goeb M. (2008) “Modèles Géométriques et Mesures de cercles contraints”, Ph. D. Thesis, University of Saint-Etienne, FranceGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.CPE LyonVilleurbanne CedexFrance
  2. 2.Université de Lyon /CNRS, UMR 5516, Laboratoire Hubert Curien /Université de Saint-Etienne, Jean MonnetSaint-EtienneFrance

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