On geodesic properties of polygons relevant to linear time triangulation

Triangulating a simple polygon ofn vertices inO(n) time is one of the main open problems in computational geometry. The fastest algorithm to date, due to Tarjan and van Wyk, runs inO(n log logn), but several classes of simple polygons have been shown to admit linear time traingulation. Famous examples of such classes are: star-shaped, monotone, spiral, edge visible, and weakly externally visible polygons. The notion of geodesic paths is used here to characterize all classes of polygons for which linear time triangulation algorithms are known. First we introduce a new class of polygons,palm polygons, which subsumes many known classes of polygons for which linear time triangulation algorithms exist, and present a linear time algorithm for triangulating polygons in this class. Then a class of polygons,crab polygons, is defined and shown to contain all classes of existing polygons for which linear time triangulation algorithms are known. As a byproduct of this characterization, a new, very simple linear time algorithm for triangulating star-shaped polygons is obtained.

[1]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[2]  Kyung-Yong Chwa,et al.  A new triangulation-linear class of simple polygons , 1987 .

[3]  D. T. Lee,et al.  Visibility of a simple polygon , 1983, Comput. Vis. Graph. Image Process..

[4]  Godfried T. Toussaint.  Shortest Path Solves Translation Separability of Polygons , 1986, IAS.

[5]  Godfried T. Toussaint.  A new linear algorithm for triangulating monotone polygons , 1984, Pattern Recognit. Lett..

[6]  Godfried T. Toussaint,et al.  An Optimal Algorithm for Determining the Visibility of a Polygon from an Edge , 1981, IEEE Transactions on Computers.

[7]  Sung Yong Shin,et al.  A Linear Time Algorithm for Triangulating a Point-Visible Polygon , 1985, ACM Trans. Graph..

[8]  Robert E. Tarjan,et al.  An O(n log log n)-Time Algorithm for Triangulating a Simple Polygon , 1988, SIAM J. Comput..

[9]  Godfried T. Toussaint,et al.  On a convex hull algorithm for polygons and its application to triangulation problems , 1982, Pattern Recognit..

[10]  Godfried T. Toussaint,et al.  Applications of a two-dimensional hidden-line algorithm to other geometric problems , 2005, Computing.

[11]  David Avis,et al.  A Linear Algorithm for Computing the Visibility Polygon from a Point , 1981, J. Algorithms.

[12]  D. T. Lee,et al.  An Optimal Algorithm for Finding the Kernel of a Polygon , 1979, JACM.

[13]  J. van Leeuwen,et al.  Triangulating a starshaped polygon , 1980 .

[14]  Bernard Chazelle,et al.  Triangulation and shape-complexity , 1984, TOGS.