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Intersections and circuits in sets of line segments

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Abstract

Sets of straight line segments with special structures and properties appear in various applications of geometric modeling, such as scientific visualization, computer-aided design, and medical image processing. In this paper, we derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. In particular, we consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets.

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Notes

  1. A graph is maximal outerplanar if it has a plane embedding in which all vertices belong to the outer face, and adding any edge to the graph causes it to no longer have this property.

  2. We use this nomenclature instead of triangle-free graph in order to avoid confusion between geometric and graph theoretic triangles.

  3. The nomenclature is derived from Buffon’s Needle Problem, which investigates the probability that a needle will fall on a line when dropped on a sheet with equally spaced lines.

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Acknowledgements

We thank two anonymous referees for their helpful suggestions which helped improve the paper. We thank Edinah Gnang for suggesting the study of Buffon segments and Stephen Hartke for several useful discussions. This work is partially supported by NSF-DMS Grants 1603823 and 1604458.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Slippery Rock University, Slippery Rock, PA, 16057, USA

    Boris Brimkov

  2. Department of Mathematics, San Jose State University, San Jose, CA, 95192, USA

    Jesse Geneson

  3. Department of Mathematics and Computer Science, Susquehanna University, Selinsgrove, PA, 17870, USA

    Alathea Jensen

  4. Department of Mathematics and Statistics, Washington State University, Pullman, WA, 99164, USA

    Jordan Broussard

  5. Broad Institute of MIT and Harvard, Cambridge, MA, 02142, USA

    Pouria Salehi Nowbandegani

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  1. Boris Brimkov
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  2. Jesse Geneson
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  3. Alathea Jensen
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  4. Jordan Broussard
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  5. Pouria Salehi Nowbandegani
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Correspondence to Boris Brimkov.

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Brimkov, B., Geneson, J., Jensen, A. et al. Intersections and circuits in sets of line segments. J Comb Optim 44, 2302–2323 (2022). https://doi.org/10.1007/s10878-021-00731-3

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