On the minimality of polygon triangulation
- 36 Downloads
The problem of triangulating a polygon using the minimum number of triangles is treated. We show that the minimum number of triangles required to partition a simplen-gon is equal ton+2w −d − 2, wherew is the number of holes andd is the maximum number of independent degenerate triangles of then-gon. We also propose an algorithm for constructing the minimum triangulation of a simple hole-freen-gon. The algorithm takesO(nlog2n+DK2) time, whereD is the maximum number of vertices lying on the same line in then-gon andK is the number of minimally degenerate triangles of then-gon.
CR subjects classificationsG.1.6 I.1.2 I.3.5
Unable to display preview. Download preview PDF.
- 1.T. Asano, T. Asano, and Y. Ohsuga,Partitioning a polygonal region into a minimum number of triangles, The Transactions of the IECE of Japan, Vol. E 67, No. 4 (1984), pp. 232–233.Google Scholar
- 2.T. Asano, T. Asano, and R. Y. Pinter,Polygon triangulation: efficiency and minimality, Journal of Algorithms, Vol. 7, No. 2 (1986), pp. 221–231.Google Scholar
- 3.B. M. Chazelle,A theorem for polygon cutting with applications, Proceedings of the 23rd IEEE Annual Symposium on the Foundations of Computer Science (1983), pp. 339–349.Google Scholar
- 4.B. M. Chazelle and L. J. Guibas,Visibility and intersection problems in plane geometry, Discrete Computational Geometry, Vol. 4, No. 6 (1989), pp. 551–581.Google Scholar
- 5.A. Fournier and D. Y. Montuno,Triangulating simple polygons and equivalent problems, ACM Transactions on Graphics, Vol. 3, No. 2 (1984), pp. 153–174.Google Scholar
- 6.M. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan,Triangulating a simple polygon, Information Processing Letters, Vol. 7, No. 4 (1978), pp. 175–179.Google Scholar
- 7.A. Lingas,The power of non-rectilinear holes, Proceedings of the 9th Colloquium on Automata, Languages and Programming, Aarhus (1982), pp. 369–383.Google Scholar
- 8.J. O'Rourke,Art Gallery Theorems and Algorithms, Oxford University Press, Inc., New York (1987).Google Scholar
- 9.K. J. Supowit,Finding a maximum planar subset of a set of nets in a channel, IEEE Transactions on Computer-Aided Design, Vol. 6, No. 1 (1987), pp. 93–94.Google Scholar
- 10.R. E. Tarjan, and C. J. Van Wyk,An O(n loglog n)-time algorithm for triangulating a simple polygon, SIAM Journal on Computing, Vol. 17, No. 1 (1988), pp. 143–178.Google Scholar