CG International ’90 pp 443-466 | Cite as

# An Output-Complexity-Sensitive Polygon Triangulation Algorithm

## Abstract

This paper describes a new algorithm for triangulating a simple n-sided polygon. The algorithm runs in time O(n(1+t_{0})), with t_{0}< n. The quantity t_{0}measures the*shape-complexity*of the*triangulation*delivered by the algorithm. More precisely t_{0}is the number of triangles contained in the triangulation obtained that share zero edges with the input polygon and is, furthermore, related to the shape- complexity of the*input*polygon. Although the worst-case complexity of the algorithm is O(n^{2}), for several classes of polygons it runs in linear time. The practical advantages of the algorithm are that it is simple and does not require sorting or the use of balanced tree structures. On the theoretical side it is of interest because it is the first polygon triangulation algorithm the*computational*complexity of which is a function of the*output*complexity. As a side benefit we introduce a new measure of the complexity of a polygon triangulation that should find application in other contexts as well.

## Keywords

Computational Geometry Simple Polygon Left Turn Polygonal Chain Jordan Curve Theorem## Preview

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## References

- [Ca]Cairns, S. S., “An elementary proof of the Jordan-Schoenflies theorem,
*Proc. Amer. Math. Soc*., vol. 2, 1951, pp. 860–867.CrossRefMATHMathSciNetGoogle Scholar - [Ch]Chazelle, B., “A theorem on polygon cutting with applications,”
*Proc. 23rd IEEE Symposium on Foundations of Computer Science, Chicago*, November 1982.Google Scholar - [CI]Chazelle, B. and Incerpi, J., “Triangulation and shape complexity,”
*ACM Transactions on Graphics*, vol. 3, 1984, pp. 135–152.CrossRefMATHGoogle Scholar - [E1]El Gindy, H. A., “A linear algorithm for triangulating weakly externally visible polygons,” Tech. Report MS-CIS-86-75, University of Pennsylvania, September 1985.Google Scholar
- [ET1]ElGindy, H. and Toussaint, G. T., “On triangulating palm polygons in linear time,”
*Proc. Computer Graphics International ’88*, Geneva, May 24–27, 1988.Google Scholar - [ET2]ElGindy, H. and Toussaint, G. T., “On geodesic properties of polygons relevant to linear- time triangulation,”
*The Visual Computer*, vol. 5, no. 1 /2, March 1989, pp. 68–74.Google Scholar - [EAT]ElGindy, H., Avis, D. and Toussaint, G. T., “Applications of a two-dimensional hidden-line algorithm to other geometric problems,”
*Computing*, vol. 31, 1983, pp. 191–202.CrossRefMATHMathSciNetGoogle Scholar - [EET]ElGindy, H., Everett, H. and Toussaint, G. T., “Slicing an ear in linear time,” internal memorandum, School of Computer Science, McGill University.Google Scholar
- [FM]Fournier, A. and Montuno, D. Y., “Triangulating simple polygons and equivalent problems,”
*ACM Transactions on Graphics*, vol. 3, April 1984, pp. 153–174.CrossRefMATHGoogle Scholar - [FP]Feng, H-Y. F. and Pavlidis, T., “Decomposition of polygons into simpler components: feature generation for syntactic pattern recognition,”
*IEEE Transactions on Computers*, vol. C- 24, June 1975, pp. 636–650.CrossRefMATHMathSciNetGoogle Scholar - [Fo]Forder, H. G.,
*The Foundations of Euclidean Geometry*, Cambridge University Press, 1927.MATHGoogle Scholar - [GJPT]Garey, M. R., Johnson, D. S., Preparata, F. P. and Taijan, R. E., “Triangulating a simple polygon,”
*Information Processing Letters*, vol. 7, 1978, pp. 175–179.CrossRefMATHMathSciNetGoogle Scholar - [HM]Hertel, S. and Mehlhorn, K., “Fast triangulation of simple polygons,”
*Proc. FCT, LNCS*158, 1983, pp. 207–215.MathSciNetGoogle Scholar - [Ho]Ho, W.-C., “Decomposition of a polygon into triangles,”
*The Mathematical Gazette*, vol. 59, 1975, pp. 132–134.CrossRefGoogle Scholar - [KKT]Kirkpatrick, D. G., Klawe, M. M., & Tarjan, R. E., “O(n log log n) polygon triangulation with simple data structures,”
*Sixth Annual Symposium on Computational Geometry*, Berkeley, California, June 6–8, 1990.Google Scholar - [Kn1]Knopp, K.,
*Theory of Functions*,Part I, translated by F. Bagemihl from the fifth German Edition, Dover.Google Scholar - [Kn2]Knopp, K.,
*Funktionentheorie*I., Sammlung Goschen Band 668, Walter de Gruyter, 1970.Google Scholar - [LC]Lee, S. H. and Chwa, K. Y., “A new triangulation linear class of simple polygons,”
*International Journal of Computer Mathematics*, vol. 22, 1987, pp. 135–147.CrossRefMATHGoogle Scholar - [Le]Lennes, N. J., “Theorems on the simple finite polygon and polyhedron,”
*American Journal of Mathematics*, vol. 33, 1911, pp. 37–62.CrossRefMATHMathSciNetGoogle Scholar - [Le1]Levy, L. S.,
*Geometry: Modern Mathematics via the Euclidean Plane*,Prindle, Weber & Schmidt, Inc., Boston, Mass., 1970.MATHGoogle Scholar - [Ma]Mandelbrot, B. B.,
*Fractals: Form, Chance, and Dimension*, W. H. Freeman & Co., 1977.Google Scholar - [Me]Meisters, G. H., “Polygons have ears,”
*American Mathematical Monthly*, June/July 1975, pp. 648–651.Google Scholar - [RS]Rupert, J. and Seidel, R., “On the difficulty of tetrahedralizing 3-dimensional non-convex polyhedra,”
*ACM Symposium on Computational Geometry*,June 5–7 1989, Saarbrucken, West Germany, pp. 380–392.Google Scholar - [SV]Schoone, A. A. and van Leeuwen, J., “Triangulating a star-shaped polygon,” Tech. Report, RUV-CS-80-3, University of Utrecht, April 1980.Google Scholar
- [Sh1]Shermer, T., “Computing bushy and thin triangulations,” in
*Snapshots of Computational and Discrete Geometry*, G. T. Toussaint, Ed., Tech. Rept. SOCS-88.11, June 1988, pp. 119–133.Google Scholar - [Sh2]Shermer, T., “Generating anthropomorphic k-spirals,” in
*Snapshots of Computational and Discrete Geometry*, G. T. Toussaint, Ed., Tech. Rept. SOCS-88.11, June 1988, pp. 233–244.Google Scholar - [TA]Toussaint, G. T. and Avis, D., “On a convex hull algorithm for polygons and its application to triangulation problems,”
*Pattern Recognition*, vol. 15, No. 1, 1982, pp. 23–29.CrossRefMathSciNetGoogle Scholar - [To1]Toussaint, G. T., “A new linear algorithm for triangulating monotone polygons,”
*Pattern Recognition Letters*,vol. 2, March 1984, pp.Google Scholar - [To2]Toussaint, G. T., “Polygons are anthropomorphic,” Memorandum, School of Computer Science, McGill University, March, 1988.Google Scholar
- [To3]Toussaint, G. T., “New results in computational geometry relevant to pattern recognition in practice,” in
*Pattern Recognition in Practice II*, E. S. Gelsema and L. N. Kanal, Editors, North-Holland, 1986, pp. 135–146.Google Scholar - [To4]Toussaint, G. T., Editor,
*Computational Morphology*, North-Holland, 1988.MATHGoogle Scholar - [TV]Taijan, R. E. and Van Wyk, C. J., “An O(n log log n)-time algorithm for triangulating simple polygons,”
*SIAM Journal on Computing*, 1988.Google Scholar - [WS]Woo, T. C. and Shin, S. Y., “A linear time algorithm for triangulating a point-visible polygon,”
*ACM Transactions on Graphics*, vol. 4, January 1985, pp. 60–70.MATHGoogle Scholar