Numerische Mathematik

, Volume 134, Issue 2, pp 389–411 | Cite as

Far-field reflector problem and intersection of paraboloids

  • Pedro Machado Manhães de Castro
  • Quentin Mérigot
  • Boris ThibertEmail author


In this article, we propose a numerical approach to the far field reflector problem which is an inverse problem arising in geometric optics. Caffarelli et al. (Contemp Math 226:13–32, 1999) proposed an algorithm that involves the computation of the intersection of the convex hull of confocal paraboloids. We show that computing this intersection amounts to computing the intersection of a power diagram (a generalization of the Voronoi diagram) with the unit sphere. This allows us to provide an algorithm that computes efficiently the intersection of confocal paraboloids using the exact geometric computation paradigm. Furthermore, using an optimal transport formulation, we cast the far field reflector problem into a concave maximization problem. This allows us to numerically solve the far field reflector problem with up to 15k paraboloids. We also investigate other geometric optic problems that involve union of confocal paraboloids and also intersection and union of confocal ellipsoids. In all these cases, we show that the computation of these surfaces is equivalent to the computation of the intersection of a power diagram with the unit sphere.

Mathematics Subject Classification

68U05 Computer graphics; computational geometry 65K15 Mathematical programming, optimization and variational techniques  52A41 Convex functions and convex programs  35Q99 Equations of mathematical physics and other areas of application 



The authors would like to thank Dominique Attali, Olivier Devillers and Francis Lazarus for interesting discussions. Olivier Devillers suggested the approach used in the proof of the lower complexity bound for intersection of ellipsoids. P. M. M. de Castro is supported by Grant FACEPE/INRIA, APQ-0055-1.03/12. Q. Mérigot and B. Thibert would like to acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-BS01-014-01 (TOMMI) and ANR-13-BS01-0008-03 (TOPDATA) respectively.


  1. 1.
    Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 78–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski-type theorems and least-squares clustering. Algorithmica 20(1), 61–76 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertsekas, D.P., Eckstein, J.: Dual coordinate step methods for linear network flow problems. Math. Program. 42(1), 203–243 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boissonnat, J.-D., Karavelas, M.I.: On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Philadelphia, PA, USA), SODA ’03. Society for Industrial and Applied Mathematics, pp. 305–312 (2003)Google Scholar
  5. 5.
    Brónnimann, H., Burnikel, C., Pion, S.: Interval arithmetic yields efficient dynamic filters for computational geometry. Discret. Appl. Math. 109, 25–47 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L.A., Gutiérrez, C.E., Huang, Q.: On the regularity of reflector antennas. Ann. Math. Second Ser. 167(1), 299 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caffarelli, L.A., Kochengin, S., Oliker, V.I.: On the numerical solution of the problem of reflector design with given far-field scattering data. Contemp. Math. 226, 13–32 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L.A., Oliker, V.I.: Weak solutions of one inverse problem in geometric optics. J. Math. Sci. 154(1), 39–49 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cgal, Computational Geometry Algorithms Library.
  10. 10.
    Delage, C.: CGAL-based first prototype implementation of Möbius diagram in 2D. Technical Report ECG-TR-241208-01, INRIA Sophia-Antipolis, 2003Google Scholar
  11. 11.
    Glimm, T., Oliker, V.: Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem. J. Math. Sci. 117(3), 4096–4108 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guan, P., Wang, X.-J.: On a Monge–Ampere equation arising in geometric optics. J. Differ. Geom. 48(2), 205–223 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kettner, L., Mehlhorn, K., Schirra, S., Yap, C.K.: Classroom examples of robustness problems in geometric computations. Comput. Geom. Theory Appl. 40, 61–79 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kitagawa, J.: An iterative scheme for solving the optimal transportation problem (2012) (preprint). arXiv:1208.5172
  15. 15.
    Kochengin, S.A., Oliker, V.I.: Determination of reflector surfaces from near-field scattering data. Inverse Probl. 13(2), 363 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mérigot, Q.: A multiscale approach to optimal transport. In: Computer Graphics Forum, vol. 30, pp. 1583–1592. Wiley Online Library, New York (2011)Google Scholar
  17. 17.
    Mulmuley, K. (ed.): Computational geometry—an introduction through randomized algorithms. Prentice Hall, Englewood Cliffs (1994)zbMATHGoogle Scholar
  18. 18.
    Oliker, V.I.: Mathematical aspects of design of beam shaping surfaces in geometrical optics. In: Trends in Nonlinear Analysis, pp. 193–224 (2003)Google Scholar
  19. 19.
    Oliker, V.I.: A rigorous method for synthesis of offset shaped reflector antennas. Comput. Lett. 2(1–2), 1–2 (2006)Google Scholar
  20. 20.
    Oliker, V.I.: A characterization of revolution quadrics by a system of partial differential equations. Proc. Am. Math. Soc. 138(11), 4075–4080 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sack, J.-R., Urrutia, J. (eds.): Handbook of Computational Geometry. North-Holland Publishing Co., Amsterdam (2000)zbMATHGoogle Scholar
  22. 22.
    Wang, X.J.: On the design of a reflector antenna ii. Calc. Var. Partial Differ. Equ. 20(3), 329–341 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, X.-J.: On the design of a reflector antenna. Inverse Probl. 12(3), 351 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Pedro Machado Manhães de Castro
    • 1
  • Quentin Mérigot
    • 2
    • 3
  • Boris Thibert
    • 2
    • 3
    Email author
  1. 1.Centro de InformáticaUniversidade Federal de PernambucoRecifeBrazil
  2. 2.Université Grenoble Alpes, LJKGrenobleFrance
  3. 3.CNRS, LJKGrenobleFrance

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