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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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