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Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications

Authors Sándor P. Fekete , Prahlad Narasimhan Kasthurirangan , Phillip Keldenich , Fabian Kollhoff , Chek-Manh Loi , Michael Perk

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LIPIcs.SoCG.2026.45.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • General and reference → Experimentation
  • Mathematics of computing → Arbitrary-precision arithmetic
Keywords
  • Visibility
  • line segments
  • link distance
  • window partition
  • computation
  • implementation
  • robustness
  • scalability
  • exactness
  • CGAL

Metrics

Abstract

The weak visibility polygon of a line segment s inside a simple polygon P, denoted by V_P(s), is the region of the polygon that is visible from at least one point on s. Given its fundamental nature in computational geometry, several algorithms have been proposed to compute weak visibility polygons efficiently, each with different trade-offs in terms of preprocessing time, query time, and space complexity. Although there are many applications that require computing these polygons such as computer graphics, robot motion planning, and network communication systems, there is a lack of any implementations of these algorithms in the literature - not to mention one that is exact, robust, and scalable. Furthermore, weak segment visibility polygons are used as basic building blocks in several other algorithms, such as in minimum-link path computation.
In this work, we present an implementation of an optimal linear-time algorithm for computing the weak visibility polygon of a segment inside a triangulated simple polygon. Our implementation provides exact, robust geometric primitives and optimizations to handle large inputs with more than 18,000,000 vertices. We demonstrate two concrete applications: (1) construction of window partitions, a standard data structure in visibility algorithms, and (2) support for optimal minimum-link path queries between two points in a simple polygon, the latter serving as a direct use case of the former. Experimental results on a variety of polygon families confirm that the end-to-end running time scales linearly with the size of the polygon and is dominated by the cost of computing the triangulation, validating the practicality and scalability of the approach. The implementation is released as open source in the format of a CGAL package to support reproducibility and further research.

Cite As Get BibTex

Sándor P. Fekete, Prahlad Narasimhan Kasthurirangan, Phillip Keldenich, Fabian Kollhoff, Chek-Manh Loi, and Michael Perk. Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.45

Author Details

Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
  • L3S Research Center, Hannover, Germany
Prahlad Narasimhan Kasthurirangan
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA
Phillip Keldenich
  • Department of Computer Science, TU Braunschweig, Germany
Fabian Kollhoff
  • Department of Computer Science, TU Braunschweig, Germany
Chek-Manh Loi
  • Department of Computer Science, TU Braunschweig, Germany
Michael Perk
  • Department of Computer Science, TU Braunschweig, Germany

Funding

Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) as part of project Computational Geometry: Solving Hard Optimization Problems (CG:SHOP) - 444569951.
  • Kasthurirangan, Prahlad Narasimhan: Work was carried out during a research stay at TU Braunschweig. Partially supported by the US Office of Naval Research under grant no. N00014-26-1-2088.

Acknowledgements

We thank the authors of [Baum et al., 2018] for sharing parts of their implementation with us.

Supplementary Materials

References

  1. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. The art gallery problem is ∃ℝ-complete. Journal of the ACM, 69(1):1-70, 2021. URL: https://doi.org/10.1145/3486220.
  2. Esther M. Arkin, Joseph S. B. Mitchell, and Subhash Suri. Optimal link path queries in a simple polygon. In Symposium on Discrete Algorithms (SODA), pages 269-279, 1992. URL: http://dl.acm.org/citation.cfm?id=139404.139462.
  3. Boris Aronov, Leonidas J. Guibas, Marek Teichmann, and Li Zhang. Visibility queries and maintenance in simple polygons. Discrete and Computational Geometry, 27(4):461-483, 2002. URL: https://doi.org/10.1007/S00454-001-0089-9.
  4. Moritz Baum, Thomas Bläsius, Andreas Gemsa, Ignaz Rutter, and Franziska Wegner. Scalable exact visualization of isocontours in road networks via minimum-link paths. Journal of Computational Geometry, 9(1):27-73, 2018. URL: https://doi.org/10.20382/JOCG.V9I1A2.
  5. Jean-Daniel Boissonnat, Olivier Devillers, Monique Teillaud, and Mariette Yvinec. Triangulations in CGAL. In Symposium on Compuational Geometry (SoCG), pages 11-18, 2000. URL: https://doi.org/10.1145/336154.336165.
  6. Prosenjit Bose, Anna Lubiw, and J. Ian Munro. Efficient visibility queries in simple polygons. Computational Geometry, 23(3):313-335, 2002. URL: https://doi.org/10.1016/S0925-7721(01)00070-0.
  7. Reilly Browne, Prahlad Narasimham Kasthurirangan, Joseph S. B. Mitchell, and Valentin Polishchuk. Constant-factor approximation algorithms for convex cover and hidden set in a simple polygon. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1357-1365, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00083.
  8. Francisc Bungiu, Michael Hemmer, John Hershberger, Kan Huang, and Alexander Kröller. Efficient computation of visibility polygons. CoRR, abs/1403.3905, 2014. URL: https://arxiv.org/abs/1403.3905.
  9. Mojtaba Nouri Bygi and Mohammad Ghodsi. Weak visibility queries in simple polygons. In Canadian Conference on Computational Geometry (CCCG), 2011. URL: http://www.cccg.ca/proceedings/2011/papers/paper95.pdf.
  10. Mojtaba Nouri Bygi and Mohammad Ghodsi. Weak visibility queries of line segments in simple polygons and polygonal domains. CoRR, abs/1310.7197, 2013. URL: https://arxiv.org/abs/1310.7197.
  11. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete and Computational Geometry, 6:485-524, 1991. URL: https://doi.org/10.1007/BF02574703.
  12. Bernard Chazelle. The computational geometry impact task force report: An executive summary. In Workshop on Applied Computational Geometry (WACG), pages 59-65, 1996. URL: https://doi.org/10.1007/BFB0014485.
  13. Danny Z. Chen and Haitao Wang. Weak visibility queries of line segments in simple polygons. Computational Geometry, 48(6):443-452, 2015. URL: https://doi.org/10.1016/j.comgeo.2015.02.001.
  14. Mark de Berg. On rectilinear link distance. Computational Geometry, 1:13-34, 1991. URL: https://doi.org/10.1016/0925-7721(91)90010-C.
  15. Mark De Berg, Otfried Cheong, Marc Van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  16. Pedro J de Rezende, Cid C de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C Tozoni. Engineering art galleries. In Algorithm Engineering: Selected Results and Surveys, pages 379-417. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-49487-6_12.
  17. Nakju Lett Doh, Chanki Kim, and Wan Kyun Chung. A practical path planner for the robotic vacuum cleaner in rectilinear environments. IEEE Transactions on Consumer Electronics, 53(2):519-527, 2007. URL: https://doi.org/10.1109/TCE.2007.381724.
  18. Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph SB Mitchell. Touring a sequence of polygons. In Symposium on the Theory of Computing (STOC), pages 473-482, 2003. Google Scholar
  19. Günther Eder, Martin Held, Steinþór Jasonarson, Philipp Mayer, and Peter Palfrader. Salzburg database of polygonal data: Polygons and their generators. Data in Brief, 31:105984, 2020. URL: https://doi.org/10.1016/j.dib.2020.105984.
  20. Robert Fitch, Zack Butler, and Daniela Rus. 3d rectilinear motion planning with minimum bend paths. In Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No. 01CH37180), volume 3, pages 1491-1498. IEEE, 2001. URL: https://doi.org/10.1109/IROS.2001.977191.
  21. Submir Kumar Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007. Google Scholar
  22. Andrew Gibiansky. Finger trees, 2014. URL: https://andrew.gibiansky.com/blog/haskell/finger-trees/.
  23. Ronald L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132-133, 1972. URL: https://doi.org/10.1016/0020-0190(72)90045-2.
  24. Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert Endre Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209-233, 1987. URL: https://doi.org/10.1007/BF01840360.
  25. Leonidas J. Guibas, John Hershberger, Joseph S. B. Mitchell, and Jack Snoeyink. Approximating polygons and subdivisions with minimum link paths. International Journal of Computational Geometry and Applications, 3(4):383-415, 1993. URL: https://doi.org/10.1142/S0218195993000257.
  26. Leonidas J. Guibas, Edward M. McCreight, Michael F. Plass, and Janet R. Roberts. A new representation for linear lists. In Symposium on the Theory of Computing (STOC), pages 49-60, 1977. URL: https://doi.org/10.1145/800105.803395.
  27. Mart Hagedoorn and Valentin Polishchuk. Link diameter, radius and 2-point link distance queries in polygonal domains. In Workshop on Algorithms and Data Structures (WADS), pages 34:1-34:15, 2025. ISSN: 1868-8969. URL: https://doi.org/10.4230/LIPIcs.WADS.2025.34.
  28. Michael Hemmer, Kan Huang, Francisc Bungiu, and Ning Xu. 2D visibility computation. In CGAL User and Reference Manual. CGAL Editorial Board, 6.1 edition, 2025. URL: https://doc.cgal.org/6.1/Manual/packages.html#PkgVisibility2.
  29. Ralf Hinze and Ross Paterson. Finger trees: a simple general-purpose data structure. Journal of Functional Programming, 16(2):197-217, 2006. URL: https://doi.org/10.1017/S0956796805005769.
  30. Scott Huddleston and Kurt Mehlhorn. A new data structure for representing sorted lists. Acta informatica, 17(2):157-184, 1982. URL: https://doi.org/10.1007/BF00288968.
  31. Hiroshi Imai and Masao Iri. Polygonal approximations of a curve—formulations and algorithms. In Machine Intelligence and Pattern Recognition, volume 6, pages 71-86. Elsevier, 1988. Google Scholar
  32. Phillip-Raphaël Keldenich, Fabian Kollhoff, Michael Perk, Chek-Manh Loi, and Prahlad Narasimhan Kasthurirangan. Line segment visibility in simple polygons, March 2026. URL: https://doi.org/10.5281/zenodo.19194126.
  33. Lutz Kettner, Kurt Mehlhorn, Sylvain Pion, Stefan Schirra, and Chee Yap. Classroom examples of robustness problems in geometric computations. Computational Geometry, 40(1):61-78, 2008. URL: https://doi.org/10.1016/J.COMGEO.2007.06.003.
  34. K. Kolarov and B. Roth. On the number of links and placement of telescoping manipulators in an environment with obstacles. In International Conference on Advanced Robotics (ICAR), pages 988-993 vol.2, 1991. URL: https://doi.org/10.1109/ICAR.1991.240543.
  35. Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk, and Frank Staals. On the complexity of minimum-link path problems. Journal of Computational Geometry, 8(2):80-108, March 2017. URL: https://doi.org/10.20382/jocg.v8i2a5.
  36. Alexander Kröller, Tobias Baumgartner, Sándor P Fekete, and Christiane Schmidt. Exact solutions and bounds for general art gallery problems. Journal of Experimental Algorithmics, 17:2-1, 2012. URL: https://doi.org/10.1145/2133803.2184449.
  37. D. T. Lee and Franco P. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14(3):393-410, 1984. URL: https://doi.org/10.1002/NET.3230140304.
  38. Jyh-Ming Lien and Nancy M. Amato. Approximate convex decomposition of polygons. Computational Geometry, 35(1):100-123, 2006. URL: https://doi.org/10.1016/j.comgeo.2005.10.005.
  39. Anil Maheshwari, Jörg-Rüdiger Sack, and Hristo N. Djidjev. Link distance problems. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 519-558. North Holland / Elsevier, 2000. URL: https://doi.org/10.1016/B978-044482537-7/50013-9.
  40. Joseph S. B. Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-link paths revisited. Computational Geometry, 47(6):651-667, 2014. Special issue on EuroCG'11. URL: https://doi.org/10.1016/J.COMGEO.2013.12.005.
  41. Joseph S. B. Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-link paths revisited. Computational Geometry, 47(6):651-667, August 2014. URL: https://doi.org/10.1016/j.comgeo.2013.12.005.
  42. Joseph S. B. Mitchell, Günter Rote, and Gerhard J. Woeginger. Minimum-link paths among obstacles in the plane. In Symposium on Computational Geometry (SoCG), pages 63-72. ACM, 1990. URL: https://doi.org/10.1145/98524.98537.
  43. Joseph S. B. Mitchell and Subhash Suri. Separation and approximation of polyhedral objects. Computational Geometry, 5:95-114, 1995. URL: https://doi.org/10.1016/0925-7721(95)00006-U.
  44. Joseph SB Mitchell. Shortest paths and networks. In Handbook of Discrete and Computational Geometry, pages 445-466. CRC Publishing, 1997. Google Scholar
  45. Joseph SB Mitchell et al. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, volume 334, pages 633-702. North Holland, 2000. URL: https://doi.org/10.1016/B978-044482537-7/50016-4.
  46. Joseph O'Rourke. Art gallery theorems and algorithms. Oxford University Press, Inc., USA, September 1987. Google Scholar
  47. John H. Reif and James A. Storer. Minimizing turns for discrete movement in the interior of a polygon. IEEE Journal on Robotics and Automation, 3(3):182-193, 1987. URL: https://doi.org/10.1109/JRA.1987.1087092.
  48. Jörg-Rüdiger Sack and Jorge Urrutia. Handbook of computational geometry. Elsevier, 1999. Google Scholar
  49. Subhash Suri. Minimum link paths in polygons and related problems. PhD thesis, The Johns Hopkins University, 1987. Google Scholar
  50. Subhash Suri. On some link distance problems in a simple polygon. IEEE Transactions on Robotics and Automation, 6(1):108-113, 1990. URL: https://doi.org/10.1109/70.88124.
  51. Subhash Suri and Joseph O'Rourke. Worst-case optimal algorithms for constructing visibility polygons with holes. In Alok Aggarwal, editor, Proceedings of the Second Annual ACM SIGACT/SIGGRAPH Symposium on Computational Geometry, Yorktown Heights, NY, USA, June 2-4, 1986, pages 14-23. ACM, 1986. URL: https://doi.org/10.1145/10515.10517.
  52. Ichiro Suzuki and Masafumi Yamashita. Searching for a mobile intruder in a polygonal region. SIAM Journal on Computing, 21(5):863-888, 1992. URL: https://doi.org/10.1137/0221051.
  53. Mikko Sysikaski. Rectilinear minimum link paths in two and higher dimensions. Master’s thesis, University of Helsinki, 2019. Available at URL: https://helda.helsinki.fi/bitstreams/9c8e7b5f-f4ac-42ca-bbc6-bb39a4b24782/download.
  54. Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing, 16(3):421-444, 1987. URL: https://doi.org/10.1137/0216030.
  55. The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 6.1 edition, 2025. URL: https://doc.cgal.org/6.1/Manual/packages.html.
  56. Csaba D Toth, Joseph O'Rourke, and Jacob E Goodman. Handbook of discrete and computational geometry. CRC press, 2017. Google Scholar
  57. David Phillip Wagner. Path planning algorithms under the link-distance metric. Phd thesis, Dartmouth College, 2006. Available at URL: https://digitalcommons.dartmouth.edu/cgi/viewcontent.cgi?article=1015&context=dissertations.
  58. Hu Xu, Lei Shu, and May Huang. Planning paths with fewer turns on grid maps. In Proceedings of the International Symposium on Combinatorial Search, volume 4, pages 193-200, 2013. URL: https://doi.org/10.1609/SOCS.V4I1.18298.
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