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Minkowski-Lorentz Spaces Applications: Resolution of Apollonius and Dupin Problems

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Encyclopedia of Computer Graphics and Games

Synonyms

Linear solving of Apollonius problem; Linear solving of Dupin problem

Definitions

Introduction of Minkowski-Lorentz spaces to simplify Euclidean 2- or 3-dimensional problems.

Introduction

The Apollonius problem is the determination, in a plane, of a circle tangent to three given geometric circles which corresponds to 23 = 8 Apollonius cases using oriented circles. The use of the Minkowski-Lorentz space permits to choice one over eight cases. Moreover, it generalizes when an oriented circle becomes an oriented line, the new number of solutions is given using the same algorithm whereas the new number of solutions is multiplied by 4 in the usual Euclidean affine space. The centers of the circles tangent to two given circles are on a conic. So, solving one of the Apollonius problems (with oriented circles) leads to the computation of the intersection of two conics whereas the problem is linear on the circles space Λ3 in the Minkowski-Lorentz space L3,1. The Dupin problem is the...

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Authors and Affiliations

  1. L.I.B. University of Burgundy-Franche-Comté, Dijon, France

    L. Garnier & L. Druoton

  2. LAMAV-CGAO, Valenciennes, France

    J. P. Bécar

  3. LIM-SIC, UMR CNRS 7252, Université de Poitiers, Chasseneuil, France

    L. Fuchs

  4. Laboratoire IRIT, UMR CNRS 5505, Université Paul Sabatier, Toulouse, France

    G. Morin

Authors
  1. L. Garnier
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  2. J. P. Bécar
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  3. L. Druoton
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  4. L. Fuchs
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  5. G. Morin
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Corresponding author

Correspondence to L. Garnier .

Editor information

Editors and Affiliations

  1. Inst. for Ed., Research, a. Scholarships, Los Angeles, CA, USA

    Newton Lee

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Garnier, L., Bécar, J.P., Druoton, L., Fuchs, L., Morin, G. (2019). Minkowski-Lorentz Spaces Applications: Resolution of Apollonius and Dupin Problems. In: Lee, N. (eds) Encyclopedia of Computer Graphics and Games. Springer, Cham. https://doi.org/10.1007/978-3-319-08234-9_381-1

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  • DOI: https://doi.org/10.1007/978-3-319-08234-9_381-1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08234-9

  • Online ISBN: 978-3-319-08234-9

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