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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References
Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \mathbb{R} \)-complete. J. ACM 69(1), 1–70 (2022). https://doi.org/10.1145/3188745.3188868
Below, A., Brehm, U., De Loera, J.A., Richter-Gebert, J.: Minimal simplicial dissections and triangulations of convex 3-polytopes. Discrete Comput. Geom. 24(1), 35–48 (2000). https://doi.org/10.1007/s004540010058
Below, A., De Loera, J.A., Richter-Gebert, J.: The complexity of finding small triangulations of convex 3-polytopes. J. Algorithms 50(2), 134–167 (2004). https://doi.org/10.1016/S0196-6774(03)00092-0
Benbernou, N. M., Demaine, E. D., Demaine, M. L., Kurdia, A., O’Rourke, J., Toussaint, G. T., Urrutia, J., Viglietta, G.: Edge-guarding orthogonal polyhedra. In: Proceedings of the 23rd Canadian Conference on Computational Geometry (CCCG 2011), pp. 461–466, Toronto, ON (2011). http://2011.cccg.ca/PDFschedule/papers/paper50.pdf
Bezdek, A., Carrigan, B.: On nontriangulable polyhedra. Contrib. Algebra Geom. 57, 51–66 (2016). https://doi.org/10.1007/s13366-015-0248-4
Bollobás, B.: The Art of Mathematics: Coffee Time in Memphis. Cambridge University Press, Cambridge (2006). https://doi.org/10.1017/CBO9780511816574
Bonnet, É., Miltzow, T.: An approximation algorithm for the art gallery problem. In: Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77 of LIPIcs, pp. 20:1–20:15. Schloss Dagstuhl (2017). https://doi.org/10.4230/LIPIcs.SoCG.2017.20
Bygi, M.N., Daneshpajouh, S., Alipour, S., Ghodsi, M.: Weak visibility counting in simple polygons. J. Comput. Appl. Math. 288, 215–222 (2015). https://doi.org/10.1016/j.cam.2015.04.018
Cano, J., Tóth, C.D., Urrutia, J., Viglietta, G.: Edge guards for polyhedra in three-space. Comput. Geom. 104, 101859 (2022). https://doi.org/10.1016/j.comgeo.2022.101859
Chazelle, B.: Convex partitions of polyhedra: A lower bound and worst-case optimal algorithm. SIAM J. Comput. 13(3), 488–507 (1984). https://doi.org/10.1137/0213031
Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory B 18(1), 39–41 (1975). https://doi.org/10.1016/0095-8956(75)90061-1
de Berg, M.: Generalized hidden surface removal. Comput. Geom. 5(5), 249–276 (1996). https://doi.org/10.1016/0925-7721(95)00008-9
de Berg, M., Halperin, D., Overmars, M., van Kreveld, M.: Sparse arrangements and the number of views of polyhedral scenes. Int. J. Comput. Geom. Appl. 7(03), 175–195 (1997). https://doi.org/10.1142/S0218195997000120
Demouth, J., Devillers, O., Everett, H., Glisse, M., Lazard, S., Seidel, R.: On the complexity of umbra and penumbra. Comput. Geom. 42(8), 758–771 (2009). https://doi.org/10.1016/j.comgeo.2008.04.007
Grove, E.F., Murali, T.M., Vitter, J.S.: The object complexity model for hidden-surface removal. Int. J. Comput. Geom. Appl. 9(02), 207–217 (1999). https://doi.org/10.1142/S0218195999000145
Gudmundsson, J., Morin, P.: Visibility, Planar: Testing and counting. In: Proceedings of the 26th Annual Symposium on Computational Geometry (SoCG 2010), pp. 77–86. ACM Press (2010). https://doi.org/10.1145/1810959.1810973
Iwamoto, C.: Finding the minimum number of open-edge guards in an orthogonal polygon is NP-hard. IEICE Trans. Inf. Syst. 100(7), 1521–1525 (2017). https://doi.org/10.1587/transinf.2016EDL8251
Iwamoto, C., Kishi, J., Morita, K.: Lower bound of face guards of polyhedral terrains. Inf. Media Technol. 7(2), 435–437 (2012). https://doi.org/10.2197/ipsjjip.20.435
Iwamoto, C., Kitagaki, Y., Morita, K.: Finding the minimum number of face guards is NP-hard. IEICE Trans. Inf. Syst. 95(11), 2716–2719 (2012). https://doi.org/10.1587/transinf.e95.d.2716
Kahn, J., Klawe, M., Kleitman, D.: Traditional galleries require fewer watchmen. SIAM J. Algebraic Discrete Methods 4(2), 194–206 (1983). https://doi.org/10.1137/0604020
Kitsios, N., Makris, C., Sioutas, S., Tsakalidis, A.K., Tsaknakis, J., Vassiliadis, B.: An optimal algorithm for reporting visible rectangles. Inf. Process. Lett. 81(5), 283–288 (2002). https://doi.org/10.1016/S0020-0190(01)00228-9
Kokado, K., Tóth, C. D.: Nonrealizable planar and spherical occlusion diagrams. In: Abstract of the 24th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG\(^3\) 2022), pp. 60–61 (2022)
Lee, C.W., Santos, F.: Subdivisions and triangulations of polytopes. In: Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn., pp. 415–447. Chapman and Hall/CRC, Boca Raton (2017)
Lee, D.-T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986). https://doi.org/10.1109/TIT.1986.1057165
Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications, Volume 25 of Algorithms and Computation in Mathematics. Springer, Berlin (2010)
McKenna, M.: Worst-case optimal hidden-surface removal. ACM Trans. Graphics (TOG) 6(1), 19–28 (1987). https://doi.org/10.1145/27625.27627
Moet, E., Knauer, C., van Kreveld, M.: Visibility maps of segments and triangles in 3D. Comput. Geom. 39(3), 163–177 (2008). https://doi.org/10.1016/j.comgeo.2006.11.001
O’Rourke, J.: Art Gallery Theorems and Algorithms, Volume 3 of International Series of Monographs on Computer Science. Oxford University Press, New York (1987)
O’Rourke, J.: Visibility. In: Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn., pp. 875–896. Chapman and Hall/CRC, Boca Raton (2017)
Ruppert, J., Seidel, R.: On the difficulty of triangulating three-dimensional nonconvex polyhedra. Discrete Comput. Geom. 7, 227–253 (1992). https://doi.org/10.1007/BF02187840
Schönhardt, E.: Über die Zerlegung von Dreieckspolyedern in Tetraeder. Math. Ann. 98, 309–312 (1928). https://doi.org/10.1007/BF01451597
Souvaine, D. L., Veroy, R., Winslow, A.: Face guards for art galleries. In: Proceedings of the XIV Spanish Meeting on Computational Geometry, pp. 39–42, Alcalá de Henares (2011)
Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973–1027. North-Holland, Amsterdam (2000)
Viglietta, G.: Face-guarding polyhedra. Comput. Geom. 47(8), 833–846 (2014). https://doi.org/10.1016/j.comgeo.2014.04.009
Viglietta, G.: Optimally guarding 2-reflex orthogonal polyhedra by reflex edge guards. Comput. Geom. 86, 101589 (2020). https://doi.org/10.1016/j.comgeo.2019.101589
Viglietta, G.: A theory of spherical diagrams. Comput. Geom. Topol. 2(2), 2:1–2:24 (2023). https://doi.org/10.57717/cgt.v2i2.30
Acknowledgements
The authors are grateful to Joseph O’Rourke for insightful comments and suggestions that considerably improved the readability of this paper.
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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.
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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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DOI: https://doi.org/10.1007/s00373-023-02707-y