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On the Computation of Zone and Double Zone Diagrams

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Abstract

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of Asano, Matoušek and Tokuyama introduced “implicit computational geometry” in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called “a zone diagram”. The implicit nature of zone diagrams implies, as has already been observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite-dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matoušek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be “fat”).

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Acknowledgements

This paper is the output of a long and challenging iterative process. Any honest and constructive feedback that has been given to me during this process is appreciated. I want to use this opportunity to thank the editors of DCG for their patience and to thank the referee for valuable comments. I also want to thank Mario Ponce for inviting me for a visit in the Pontifical Catholic University of Chile (Santiago) during June 2015, for helpful discussions, warm hospitality, and for allowing me to speak there (and in the Pontifical Catholic University of Valparaíso) on issues related to implicit computational geometry in general and to this paper in particular. A very sad moment along the way was when I discovered that Jiří (Jirka) Matoušek, one of the founders of implicit computational geometry, passed away. Unfortunately for me, I have not had the opportunity to meet him, but at least I had the honor to communicate with him electronically and to be a coauthor of him. It was through these communications and also through his scientific outcome that I have discovered his exceptional abilities. I hope and believe that his contributions will have a lasting value and I dedicate this paper to him. Parts of this work were done during various years, when I have been associated with the following places: The Technion, Haifa, Israel (2009–2010, 2016–2017), the University of Haifa, Haifa, Israel (2010–2011), the National Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil (2011–2013), and the Institute of Mathematics and Computational Sciences (ICMC), University of São Paulo, São Carlos, Brazil (2014–2015), and it is an opportunity for me to thank the BSF and FAPESP. Special thanks are for a special postdoc fellowship (“Pós-doutorado de Excelência”) given to me when I was at IMPA.

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  1. Department of Mathematics, The Technion - Israel Institute of Technology, 3200003, Haifa, Israel

    Daniel Reem

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Dedicated to the memory of Jiří (Jirka) Matoušek (1963–2015), an outstanding scientist, one of the founders of implicit computational geometry.

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Reem, D. On the Computation of Zone and Double Zone Diagrams. Discrete Comput Geom 59, 253–292 (2018). https://doi.org/10.1007/s00454-017-9958-8

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