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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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Author Details

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

Funding

C.A was supported by MIUR Proj. "AHeAD" nsuperscript{o} 20174LF3T8. Early work of I.M. and E.P. were supported by the DACH project Voronoi++, SNF-200021E_154387. M.S. is partly supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-9/2021-14/200125), and the Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina. Early work of H.S. was partly supported by the DFG grant FE-340/11-1. C.S. was supported by project PID2019-104129GB-I00/AEI/10.13039/501100011033. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922.

Acknowledgements

Initial discussions took place at the https://dccg.upc.edu/irp2018/ in Barcelona, Spain, in 2018. We are grateful to the organizers for providing the platform to meet and collaborate.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ESA.2021.5

Abstract

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
Keywords
  • rotating rays
  • Voronoi diagram
  • oriented angular distance
  • Brocard angle
  • floodlight illumination
  • coverage problems
  • art gallery problems

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