Abstract
The chapter describes algorithms for partitioning a simple polygon into trapezoids or triangles (Seidel’s triangulation and an algorithm using up- and down-stable vertices). Chazelle’s algorithm, published in 1991 and claimed to be of linear time, is often cited as a reference, but this algorithm was never implemented; the chapter provides a brief presentation and discussion of this algorithm. This is followed by a novel procedural presentation of Mitchell’s continuous Dijkstra algorithm for subdividing the plane into a shortest-path map for supporting queries about distances to a fixed start point in the presence of polygonal obstacles.
Many are stubborn in pursuit of the path they have chosen, few in pursuit of the goal.
Friedrich Wilhelm Nietzsche (1844–1900)
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Notes
- 1.
A trapezoid is a quadrilateral with one pair of parallel sides. Thus, a trapezoid is always a convex polygon.
- 2.
Raimund Seidel is at Saarland University.
- 3.
See also the third paragraph in [16] for a detailed discussion about this.
- 4.
Addition or subtraction of indices is modulo n.
- 5.
Each node of the tree corresponds to a region of the submap. Two nodes are connected by an edge if their corresponding regions share a chord. In this case, the edge “corresponds” to the chord.
- 6.
Bernard Chazelle is at Princeton University.
- 7.
Not to be confused with wavelets in signal theory.
- 8.
Joseph S.B. Mitchell is at the State University of New York at Stony Brook.
References
Agarwal, P.K.: Ray shooting and other applications of spanning trees and low stabbing number. In: Proc. Annu. ACM Sympos. Comput. Geom., pp. 315–325 (1989)
Chazelle, B.: A theorem on polygon cutting with applications. In: Proc. Annu. Sympos. on Foundations of Computer Science, pp. 339–349 (1982)
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6, 485–524 (1991)
Chazelle, B., Incerpi, J.: Triangulation and shape-complexity. ACM Trans. Graph. 3, 135–152 (1984)
Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Hershberger, J., Sharir, M., Snoeyink, J.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12(1), 54–68 (1994)
Cheng, S.W., Janardan, R.: Space-efficient ray-shooting and intersection searching: algorithms, dynamization, and applications. In: Proc. Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. 7–16 (1991)
Clarkson, K.L., Tarjan, R.E., Wyk, C.J.V.: A fast Las Vegas algorithm for triangulating a simple polygon. Discrete Comput. Geom. 4, 423–432 (1989)
Fathauer, R.: Website of the ‘Tessellations Company’. http://members.cox.net/fathauerart/index.html (2010). Accessed July 2011
Fournier, A., Montuno, D.Y.: Triangulating simple polygons and equivalent problems. ACM Trans. Graph. 3(2), 153–174 (1984)
Garey, M.R., Johnson, D.S., Preparata, F.P., Tarjan, R.E.: Triangulating a simple polygon. Inf. Process. Lett. 7(4), 175–179 (1978)
Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: shoot a ray, take a walk. J. Algorithms 18(3), 403–431 (1995)
Hertel, S., Mehlhorn, K.: Fast triangulation of simple polygons. In: Proc. Conf. Found. Comput. Theory, pp. 207–218 (1983)
Kirkpatrick, D.G., Klawe, M.M., Tarjan, R.E.: Polygon triangulation in \({\mathcal{O}}(n \log\log n)\) time with simple data-structures. In: Proc. Annu. ACM Sympos. Comput. Geom., pp. 34–43 (1990)
Li, F., Klette, R.: Decomposing a simple polygon into trapezoids. In: Proc. CAIP. LNCS, vol. 4673, pp. 726–733 (2007)
Mitchell, J.S.B.: Shortest paths among obstacles in the plane. Int. J. Comput. Geom. Appl. 6, 309–332 (1996)
Seidel, R.: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Comput. Geom. 1, 51–64 (1991)
Sunday, D.: Algorithm 14: Tangents to and between polygons. http://softsurfer.com/Archive/algorithm_0201/ (2006). Accessed July 2011
Tarjan, R.E., Wyk, C.J.V.: An \({\mathcal{O}}(n \log\log n)\) algorithm for triangulating a simple polygon. SIAM J. Comput. 17, 143–178 (1988)
Thorup, D.: Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 3, 362–394 (1999)
Toussaint, G.T., Avis, D.: On a convex hull algorithm for polygons and its application to triangulation problems. Pattern Recognit. 15(1), 23–29 (1982)
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Li, F., Klette, R. (2011). Partitioning a Polygon or the Plane. In: Euclidean Shortest Paths. Springer, London. https://doi.org/10.1007/978-1-4471-2256-2_5
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