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Minimizing Visible Edges in Polyhedra

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Abstract

We prove that, given a polyhedron \({\mathcal {P}}\) in \({\mathbb {R}}^3\), every point in \({\mathbb {R}}^3\) that does not see any vertex of \({\mathcal {P}}\) must see eight or more edges of \({\mathcal {P}}\), and this bound is tight. More generally, this remains true if \({\mathcal {P}}\) is any finite arrangement of internally disjoint polygons in \({\mathbb {R}}^3\). We also prove that every point in \({\mathbb {R}}^3\) can see six or more edges of \({\mathcal {P}}\) (possibly only the endpoints of some these edges) and every point in the interior of \({\mathcal {P}}\) can see a positive portion of at least six edges of \({\mathcal {P}}\). These bounds are also tight.

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Acknowledgements

The authors are grateful to Joseph O’Rourke for insightful comments and suggestions that considerably improved the readability of this paper.

Funding

Research by Tóth was partially supported by NSF DMS-1800734. Research by Urrutia was partially supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM.

Author information

Authors and Affiliations

  1. Department of Mathematics, California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  2. Department of Computer Science, Tufts University, Medford, MA, USA

    Csaba D. Tóth

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Jorge Urrutia

  4. Department of Computer Science and Engineering, University of Aizu, Tsuruga, Ichimi-cho, Aizuwakamatsu-shi, Fukushima, 965-8580, Japan

    Giovanni Viglietta

Authors
  1. Csaba D. Tóth
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Giovanni Viglietta.

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A two-page extended abstract of this paper appeared in the Abstracts of the 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCGGG), pp. 70–71, Chiang Mai, 2021.

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Tóth, C.D., Urrutia, J. & Viglietta, G. Minimizing Visible Edges in Polyhedra. Graphs and Combinatorics 39, 111 (2023). https://doi.org/10.1007/s00373-023-02707-y

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