# A new algorithm for computing the convex hull of a planar point set

## Abstract

When the edges of a convex polygon are traversed along one direction, the interior of the convex polygon is always on the same side of the edges. Based on this characteristic of convex polygons, a new algorithm for computing the convex hull of a simple polygon is proposed in this paper, which is then extended to a new algorithm for computing the convex hull of a planar point set. First, the extreme points of the planar point set are found, and the subsets of point candidate for vertex of the convex hull between extreme points are obtained. Then, the ordered convex hull point sequences between extreme points are constructed separately and concatenated by removing redundant extreme points to get the convex hull. The time complexity of the new planar convex hull algorithm is O(nlogh), which is equal to the time complexity of the best output-sensitive planar convex hull algorithms. Compared with the algorithm having the same complexity, the new algorithm is much faster.

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

## References

1. Bhattacharya, B.K., Sen, S., 1997. On a simple, practical, optimal, output-sensitive randomized planar convex hull algorithm. Journal of Algorithms, 25(1):177–193. [doi:10.1006/jagm.1997.0869]

2. Brönnimann, H., Iacono, J., Katajainen, J., Morin, P., Morrison, J., Toussaint, G., 2004. Space-efficient planar convex hull algorithms. Theor. Computer Sci., 321(1):25–40. [doi:10.1016/j.tcs.2003.05.004]

3. Brönnimann, H., Chan, T.M., 2006. Space-efficient algorithms for computing the convex hull of a simple polygonal line in linear time. Comput. Geom., 34(2):75–82. [doi:10.1016/j.comgeo.2005.11.005]

4. Cui, G.H., Hong, F., Yu, X.X., 1997. A class of optimal algorithms for determine the convex hull of a set of nodes in a plane. Chin. J. Computers, 20(4):330–334 (in Chinese).

5. Jin, W.H., He, T., Liu, X.P., Tang, W.Q., Tang, R.X., 1998. A fast convex hull algorithm of planar point set based on sorted simple polygon. Chin. J. Computers, 21(6):533–539 (in Chinese).

6. Joswig, M., Ziegler, G.M., 2004. Convex hulls, oracles, and homology. J. Symb. Comput., 38:1247–1259. [doi:10.1016/j.jsc.2003.08.006]

7. Kirkpatrick, D.G., Seidel, R., 1986. The ultimate planar convex hull algorithm? SIAM J. Computers, 15(1):287–299. [doi:10.1137/0215021]

8. Kong, X.S., Cai, H.X., 1994. An algorithm for finding the convex hull of a simple polygon using active double line testing. Chin. J. Computers, 17(8):596–600 (in Chinese).

9. Lee, T.D., 1983. On finding the convex hull of a simple polygon. Int. J. Comp. & Inf., 12(2):87–98.

10. Levcopoulos, C., Lingas, A., Mitchell, J.S.B., 2002. Adaptive Algorithms for Constructing Convex Hulls and Triangulations of Polygonal Chains. 8th Scandinavian Workshop on Algorithm Theory. Turku, FL, p.80–89.

11. Liu, J.Y., 2002. Discussion on an O(n) time algorithm for the convex hull of a planar point set. Chin. J. Computers, 25(6):670–672 (in Chinese).

12. McCallum, D., Avis, D., 1979. A linear algorithm for finding the convex hull of a simple polygon. Inf. Processing Lett., 9:201–206. [doi:10.1016/0020-0190(79)90069-3]

13. Melkman, A.A., 1987. On-line construction of the convex hull of a simple polygon. Inf. Processing Lett., 25(1):11–12. [doi:10.1016/0020-0190(87)90086-X]

14. O’Rourke, J., 1998. Computational Geometry in C (2nd Ed.). Cambridge University Press, Cambridge.

15. Preparata, F.P., Shamos, M.I., 1985. Computational Geometry: An Introduction. Springer-Verlag, New York, p.45–46.

16. Sklansky, J., 1972. Measuring concavity on a rectangular mosaic. IEEE Trans. on Comput., C-21(12):1355–1364. [doi:10.1109/T-C.1972.223507]

17. Wang, Z.Q., Hong, J.Z., Xiao, L.J., 1998. An optimal real time algorithm for determine the convex hull of a set of points in a plane. Chin. J. Computers, 21(Suppl.):351–356 (in Chinese).

18. Wu, Z.H., Ye, C.Q., Pan, Y.H., 1997. An improved algorithm of convex hull computing. J. Computer-Aided Design & Computer Graphics, 9(1):9–13 (in Chinese).

19. Yao, C.A., 1981. A lower bound to finding convex hulls. J. ACM, 28(4):780–787. [doi:10.1145/322276.322289]

## Author information

### Affiliations

Authors

Project (No. 2004AA420100) supported by the National Hi-Tech Research and Development Program (863) of China

## Rights and permissions

Reprints and Permissions

Liu, Gh., Chen, Cb. A new algorithm for computing the convex hull of a planar point set. J. Zhejiang Univ. - Sci. A 8, 1210–1217 (2007). https://doi.org/10.1631/jzus.2007.A1210

• Accepted:

• Published:

• Issue Date:

### Key words

• Computational geometry
• Convex hull
• Extreme points
• Ordered convex hull point sequence

• TP391