Skip to main content

The Rectilinear Convex Hull of Line Segments

  • 80 Accesses

Part of the Lecture Notes in Computer Science book series (LNCS,volume 14292)


We explore an extension to rectilinear convexity of the classic problem of computing the convex hull of a collection of line segments. Namely, we solve the problem of computing and maintaining the rectilinear convex hull of a set of n line segments, while we simultaneously rotate the coordinate axes by an angle that goes from 0 to \(2\pi \).

We describe an algorithm that runs in optimal \(\varTheta (n\log n)\) time and \(\varTheta (n\alpha (n))\) space for segments that are non-necessarily disjoint, where \(\alpha (n)\) is the inverse of the Ackermann’s function. If instead the line segments form a simple polygonal chain, the algorithm can be adapted so as to improve the time and space complexities to \(\varTheta (n)\).


  • rectilinear convex hull
  • line segments
  • polygonal lines

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  1. 1.

    In the literature, orthogonal convexity is also known as ortho-convexity [19] or x-y convexity [14].

  2. 2.

    See the definition of the maximal r-convex hull in [15].

  3. 3.

    In the literature, \(\mathcal {O}\)-convexity is also known as D-convexity [20], Directional convexity [8], Set-theoretical D-convexity [9], or Partial convexity [13].

  4. 4.

    We remark that, since the set \(\mathcal {O}\) is formed by two orthogonal lines parallel to the coordinate axis, in this paper \(\mathcal {O}\)-convexity is equivalent to Orthogonal Convexity.


  1. Alegría, C., Orden, D., Seara, C., Urrutia, J.: Separating bichromatic point sets in the plane by restricted orientation convex hulls maintenance of maxima of 2D point sets. J. Global Optim. 85, 1–34 (2022).

    CrossRef  MATH  Google Scholar 

  2. Biedl, T., Genç, B.: Reconstructing orthogonal Polyhedra from putative vertex sets. Comput. Geomet. Theor. Appl. 44(8), 409–417 (2011).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discr. Comput. Geom. 10(4), 377–409 (1993).

  4. Davari, M.J., Edalat, A., Lieutier, A.: The convex hull of finitely generable subsets and its predicate transformer. In: 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE (2019).

  5. Devillers, O., Golin, M.J.: Incremental algorithms for finding the convex hulls of circles and the lower envelopes of parabolas. Inf. Process. Lett. 56(3), 157–164 (1995).

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Díaz-Bañez, J.M., López, M.A., Mora, M., Seara, C., Ventura, I.: Fitting a two-joint orthogonal chain to a point set. Comput. Geom. 44(3), 135–147 (2011).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Fink, E., Wood, D.: Restricted-orientation Convexity. Monographs in Theoretical Computer Science (An EATCS Series). 1st Edn. Springer, Heidelberg (2004).

  8. Franěk, V.: On algorithmic characterization of functional \(D\)-convex hulls, Ph. D. thesis, Faculty of Mathematics and Physics, Charles University in Prague (2008)

    Google Scholar 

  9. Franěk, V., Matoušek, J.: Computing \(D\)-convex hulls in the plane. Comput. Geomet. Theor. Appl. 42(1), 81–89 (2009).

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Hershberger, J.: Finding the upper envelope of \(n\) line segments in \({O}(n\log n)\) time. Inf. Process. Lett. 33, 169–174 (1989).

    CrossRef  MATH  Google Scholar 

  11. Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discr. Comput. Geomet. 1(1), 59–71 (1986).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Melkman, A.A.: On-line construction of the convex hull of a simple polyline. Inf. Process. Lett. 25(1), 11–12 (1987).

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Metelskii, N.N., Martynchik, V.N.: Partial convexity. Math. Notes 60(3), 300–305 (1996).

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Nicholl, T.M., Lee, D.T., Liao, Y.Z., Wong, C.K.: On the X-Y convex hull of a set of X-Y polygons. BIT Numer. Math. 23(4), 456–471 (1983).

    CrossRef  MATH  Google Scholar 

  15. Ottmann, T., Soisalon-Soininen, E., Wood, D.: On the definition and computation of rectilinear convex hulls. Inf. Sci. 33(3), 157–171 (1984).

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Pérez-Lantero, P., Seara, C., Urrutia, J.: A fitting problem in three dimension. In: Book of Abstracts of the XX Spanish Meeting on Computational Geometry, pp. 21–24. EGC 2023 (2023)

    Google Scholar 

  17. Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Text and Monographs in Computer Science. 1st Edn. Springer, NY (1985).

  18. Rawlins, G.J.E.: Explorations in restricted-orientation geometry, Ph. D. thesis, School of Computer Science, University of Waterloo (1987)

    Google Scholar 

  19. Rawlins, G.J., Wood, D.: Ortho-convexity and its generalizations. In: Toussaint, G.T. (ed.) Computational Morphology, Machine Intelligence and Pattern Recognition, vol. 6, pp. 137–152. North-Holland (1988).

  20. Schuierer, S., Wood, D.: Restricted-orientation visibility. Tech. Rep. 40, Institut für Informatik, Universität Freiburg (1991)

    Google Scholar 

  21. Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press (1995)

    Google Scholar 

  22. Wang, C., Zhou, R.G.: A quantum search algorithm of two-dimensional convex hull. Commun. Theoret. Phys. 73(11), 115102 (2021).

  23. Wiernik, A., Sharir, M.: Planar realizations of nonlinear Davenport-Schinzel sequences by segments. Discr. Comput. Geomet. 3(1), 15–47 (1988).

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references


Justin Dallant is supported by the French Community of Belgium via the funding of a FRIA grant. Pablo Pérez-Lantero was partially supported by project DICYT 042332PL Vicerrectoría de Investigación, Desarrollo e Innovación USACH (Chile). Carlos Seara is supported by Project PID2019-104129GB-I00/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Carlos Seara .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Alegría, C., Dallant, J., Pérez-Lantero, P., Seara, C. (2023). The Rectilinear Convex Hull of Line Segments. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43586-7

  • Online ISBN: 978-3-031-43587-4

  • eBook Packages: Computer ScienceComputer Science (R0)