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Approximating the k-Level in Three-Dimensional Plane Arrangements

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Abstract

Let H be a set of n non-vertical planes in three dimensions, and let r < n be a parameter. We give a construction that approximates the (n∕r)-level of the arrangement \(\mathcal{A}(H)\) of H by a terrain consisting of O(r∕ɛ 3) triangular faces, which lies entirely between the levels n∕r and (1 + ɛ)n∕r. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. This leads to conceptually cleaner constructions of shallow cuttings in three dimensions.

On the way, we get two other results that are of independent interest: (a) We revisit an old result of Bambah and Rogers (J Lond Math Soc 1(3):304–314, 1952) about triangulating a union of convex pseudo-disks, and provide an alternative proof that yields an efficient algorithmic implementation. (b) We provide a new construction of cuttings in two dimensions.

A preliminary version of this paper appeared in Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2016, 1193–1212 [31]. Work by Sariel Har-Peled was partially supported by NSF AF awards CCF-1421231 and CCF-1217462. Work by Haim Kaplan was partially supported by grant 1161/2011 from the German-Israeli Science Foundation, by grant 1841/14 from the Israel Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11). Work by Micha Sharir has been supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), by the Blavatnik Research Fund at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

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Notes

  1. 1.

    The term “polygonal” is somewhat misleading, as some of the boundaries of the pseudo-disks of \(\mathcal{D}\) may also be polygonal. To avoid confusion, think of the boundaries of the pseudo-disks of \(\mathcal{D}\) as smooth convex arcs (as drawn in the figures) even though they might be polygonal.

  2. 2.

    The algorithm of [34] constructs κ-divisions for a geometrically increasing sequence of values of the parameter κ, in overall O(N) time.

  3. 3.

    The paper of Chan [16] does not use shallow cuttings.

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Acknowledgements

We thank János Pach for pointing out that a variant of Theorem 2.3 is already known. We also thank the anonymous referees for their useful feedback.

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, 61801, Urbana, IL, USA

    Sariel Har-Peled

  2. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Haim Kaplan & Micha Sharir

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  1. Sariel Har-Peled
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Correspondence to Haim Kaplan .

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  1. Department of Applied Mathematics, Charles University, Praha, Czech Republic

    Martin Loebl

  2. Computer Science Institute of Charles University, Charles University, Praha, Czech Republic

    Jaroslav Nešetřil

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Robin Thomas

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Har-Peled, S., Kaplan, H., Sharir, M. (2017). Approximating the k-Level in Three-Dimensional Plane Arrangements. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_19

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