# The Rectilinear Convex Hull of Line Segments

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

## Abstract

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 $$\varTheta (n)$$.

### Keywords

• rectilinear convex hull
• line segments
• polygonal lines

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1. 1.

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

2. 2.

See the definition of the maximal r-convex hull in .

3. 3.

In the literature, $$\mathcal {O}$$-convexity is also known as D-convexity , Directional convexity , Set-theoretical D-convexity , or Partial convexity .

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.

## References

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). https://doi.org/10.1007/s10898-022-01238-9

2. Biedl, T., Genç, B.: Reconstructing orthogonal Polyhedra from putative vertex sets. Comput. Geomet. Theor. Appl. 44(8), 409–417 (2011). https://doi.org/10.1016/j.comgeo.2011.04.002

3. Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discr. Comput. Geom. 10(4), 377–409 (1993). https://doi.org/10.1007/bf02573985

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). https://doi.org/10.1109/lics.2019.8785680

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). https://doi.org/10.1016/0020-0190(95)00132-V

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). https://doi.org/10.1016/j.comgeo.2010.07.005

7. Fink, E., Wood, D.: Restricted-orientation Convexity. Monographs in Theoretical Computer Science (An EATCS Series). 1st Edn. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-18849-7

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

9. Franěk, V., Matoušek, J.: Computing $$D$$-convex hulls in the plane. Comput. Geomet. Theor. Appl. 42(1), 81–89 (2009). https://doi.org/10.1016/j.comgeo.2008.03.003

10. Hershberger, J.: Finding the upper envelope of $$n$$ line segments in $${O}(n\log n)$$ time. Inf. Process. Lett. 33, 169–174 (1989). https://doi.org/10.1016/0020-0190(89)90136-1

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). https://doi.org/10.1007/BF02187683

12. Melkman, A.A.: On-line construction of the convex hull of a simple polyline. Inf. Process. Lett. 25(1), 11–12 (1987). https://doi.org/10.1016/0020-0190(87)90086-X

13. Metelskii, N.N., Martynchik, V.N.: Partial convexity. Math. Notes 60(3), 300–305 (1996). https://doi.org/10.1007/BF02320367

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). https://doi.org/10.1007/BF01933620

15. Ottmann, T., Soisalon-Soininen, E., Wood, D.: On the definition and computation of rectilinear convex hulls. Inf. Sci. 33(3), 157–171 (1984). https://doi.org/10.1016/0020-0255(84)90025-2

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)

17. Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Text and Monographs in Computer Science. 1st Edn. Springer, NY (1985). https://doi.org/10.1007/978-1-4612-1098-6

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

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). https://doi.org/10.1016/B978-0-444-70467-2.50015-1

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

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

22. Wang, C., Zhou, R.G.: A quantum search algorithm of two-dimensional convex hull. Commun. Theoret. Phys. 73(11), 115102 (2021). https://doi.org/10.1088/1572-9494/ac1da0

23. Wiernik, A., Sharir, M.: Planar realizations of nonlinear Davenport-Schinzel sequences by segments. Discr. Comput. Geomet. 3(1), 15–47 (1988). https://doi.org/10.1007/BF02187894

## Acknowledgements

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.

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Correspondence to Carlos Seara .

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