The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination

Authors Carlos Alegría , Ioannis Mantas , Evanthia Papadopoulou , Marko Savić , Hendrik Schrezenmaier, Carlos Seara, Martin Suderland

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Carlos Alegría
  • Dipartimento di Ingegneria, Università Roma Tre, Rome, Italy
Ioannis Mantas
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Marko Savić
  • Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia
Hendrik Schrezenmaier
  • Institut für Mathematik, Technische Universität Berlin, Germany
Carlos Seara
  • Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland


Initial discussions took place at the in Barcelona, Spain, in 2018. We are grateful to the organizers for providing the platform to meet and collaborate.

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Carlos Alegría, Ioannis Mantas, Evanthia Papadopoulou, Marko Savić, Hendrik Schrezenmaier, Carlos Seara, and Martin Suderland. The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays, and the distance function is the counterclockwise angular distance between a point and a ray-site. This novel Voronoi diagram is motivated by illumination and coverage problems, where a domain has to be covered by floodlights (wedges) of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the ray Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle. This last algorithm improves upon previous results, settling an interesting open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • rotating rays
  • Voronoi diagram
  • oriented angular distance
  • Brocard angle
  • floodlight illumination
  • coverage problems
  • art gallery problems


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  1. Carlos Alegría-Galicia, David Orden, Carlos Seara, and Jorge Urrutia. Illuminating polygons by edge-aligned floodlights of uniform angle (Brocard illumination). In Proc. European Workshop on Computational Geometry, pages 281-284, 2017. Google Scholar
  2. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. URL:
  3. Piotr Berman, Jieun Jeong, Shiva P. Kasiviswanathan, and Bhuvan Urgaonkar. Packing to angles and sectors. In Proc. ACM symposium on Parallel algorithms and architectures, pages 171-180, 2007. Google Scholar
  4. Arthur Bernhart. Polygons of pursuit. Scripta Mathematica, 24(1):23-50, 1959. Google Scholar
  5. Ádám Besenyei. The Brocard angle and a geometrical gem from Dmitriev and Dynkin. The American Mathematical Monthly, 122(5):495-499, 2015. Google Scholar
  6. Prosenjit Bose, Leonidas J. Guibas, Anna Lubiw, Mark Overmars, Diane Souvaine, and Jorge Urrutia. The floodlight problem. International Journal of Computational Geometry & Applications, 7(1-2):153-163, 1997. Google Scholar
  7. John Casey. A sequel to the first six books of the elements of Euclid. Dublin University Press, 1888. Google Scholar
  8. Felipe Contreras, Jurek Czyzowicz, Nicolas Fraiji, and Jorge Urrutia. Illuminating triangles and quadrilaterals with vertex floodlights. In Proc. Canadian Conference on Computational Geometry, 1998. Google Scholar
  9. Jurek Czyzowicz, Stefan Dobrev, Benson Joeris, Evangelos Kranakis, Danny Krizanc, Jan Maňuch, Oscar Morales-Ponce, Jaroslav Opatrny, Ladislav Stacho, and Jorge Urrutia. Monitoring the plane with rotating radars. Graphs and Combinatorics, 31(2):393-405, 2015. Google Scholar
  10. Jurek Czyzowicz, Eduardo Rivera-Campo, and Jorge Urrutia. Optimal floodlight illumination of stages. In Proc. Canadian Conference on Computational Geometry, pages 393-398, 1993. Google Scholar
  11. Mark de Berg, Joachim Gudmundsson, Herman Haverkort, and Michael Horton. Voronoi diagrams with rotational distance costs. In Computational Geometry Week: Young Researchers Forum, 2017. Google Scholar
  12. Nikolai A. Dmitriev and E Dynkin. On characteristic roots of stochastic matrices. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 10(2):167-184, 1946. Google Scholar
  13. Vladimir Estivill-Castro, Joseph O'Rourke, Jorge Urrutia, and Dianna Xu. Illumination of polygons with vertex lights. Information Processing Letters, 56(1):9-13, 1995. Google Scholar
  14. Sergiu Hart and Micha Sharir. Nonlinearity of Davenport—Schinzel sequences and of generalized path compression schemes. Combinatorica, 6(2):151-177, 1986. Google Scholar
  15. John Hershberger. Finding the upper envelope of n line segments in O(nlog n) time. Information Processing Letters, 33(4):169-174, 1989. Google Scholar
  16. Dan Ismailescu. Illuminating a convex polygon with vertex lights. Periodica Mathematica Hungarica, 57(2):177-184, 2008. Google Scholar
  17. Rolf Klein. Concrete and Abstract Voronoi diagrams, volume 400. Springer Science & Business Media, 1989. Google Scholar
  18. Rolf Klein, Elmar Langetepe, and Zahra Nilforoushan. Abstract Voronoi diagrams revisited. Computational Geometry: Theory and Applications, 42(9):885-902, 2009. Google Scholar
  19. Rolf Klein and Andrzej Lingas. Hamiltonian abstract Voronoi diagrams in linear time. In Proc. International Symposium on Algorithms and Computation, pages 11-19, 1994. Google Scholar
  20. Evangelos Kranakis, Danny Krizanc, and Oscar Morales. Maintaining connectivity in sensor networks using directional antennae. In Theoretical Aspects of Distributed Computing in Sensor Networks, pages 59-84. Springer, 2011. Google Scholar
  21. Azin Neishaboori, Ahmed Saeed, Khaled A. Harras, and Amr Mohamed. On target coverage in mobile visual sensor networks. In Proc. ACM International Symposium on Mobility Management and Wireless Access, pages 39-46, 2014. Google Scholar
  22. Joseph O'Rourke. Visibility. In Handbook of discrete and computational geometry, pages 875-896. CRC Press, 2017. Google Scholar
  23. Joseph O'Rourke, Thomas Shermer, and Ileana Streinu. Illuminating convex polygons with vertex floodlights. In Proc. Canadian Conference on Computational Geometry, pages 151-156, 1995. Google Scholar
  24. Micha Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. Discrete & Computational Geometry, 12(3):327-345, 1994. Google Scholar
  25. William Steiger and Ileana Streinu. Illumination by floodlights. Computational Geometry, 10(1):57-70, 1998. Google Scholar
  26. Tsuyoshi Taki, Jun-ichi Hasegawa, and Teruo Fukumura. Development of motion analysis system for quantitative evaluation of teamwork in soccer games. In Proc. IEEE International Conference on Image Processing, volume 3, pages 815-818. IEEE, 1996. Google Scholar
  27. Csaba D. Tóth. Art galleries with guards of uniform range of vision. Computational Geometry, 21(3):185-192, 2002. Google Scholar
  28. Jorge Urrutia. Art gallery and illumination problems. In Handbook of Computational Geometry, pages 973-1027. Elsevier, 2000. Google Scholar
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