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Breaking a polygon into monotone polygons. A simple polygon may be easily cut into monotone polygons in O(n log n) time. However, since a triangle is a monotone polygon, polygon triangulation is in fact cutting a polygon into monotone ones, and it may be performed for simple polygons in O(n) time with a complex algorithm. [6]
A monotone polygon can be split into two monotone chains. A polygon that is monotone with respect to the y-axis is called y-monotone. A monotone polygon with n vertices can be triangulated in O(n) time. Assuming a given polygon is y-monotone, the greedy algorithm begins by walking on one chain of the polygon from top to bottom while adding ...
It has 10 chapters, whose topics include the original art gallery theorem and Fisk's triangulation-based proof; rectilinear polygons; guards that can patrol a line segment rather than a single point; special classes of polygons including star-shaped polygons, spiral polygons, and monotone polygons; non-simple polygons; prison yard problems, in ...
In decision problem versions of the art gallery problem, one is given as input both a polygon and a number k, and must determine whether the polygon can be guarded with k or fewer guards. This problem is ∃ R {\displaystyle \exists \mathbb {R} } -complete , as is the version where the guards are restricted to the edges of the polygon. [ 10 ]
The convex hull of a simple polygon can also be found in linear time, faster than algorithms for finding convex hulls of points that have not been connected into a polygon. [6] Constructing a triangulation of a simple polygon can also be performed in linear time, although the algorithm is complicated.
A monotone planar subdivision with some monotone chains highlighted. A (vertical) monotone chain is a path such that the y-coordinate never increases along the path. A simple polygon is (vertical) monotone if it is formed by two monotone chains, with the first and last vertices in common. It is possible to add some edges to a planar subdivision ...
Delaunay triangulation is a completely different problem from polygon triangulation; it is a form of point set triangulation. And linear average time algorithms for Delaunay triangulation of random inputs have been known for a very long time; see e.g. Bentley, Jon Louis; Weide, Bruce W.; Yao, Andrew C. (December 1980), "Optimal Expected-Time ...
Repeatedly finding and removing a mouth from a non-convex polygon will eventually turn it into the convex hull of the initial polygon. This principle can be applied to the surrounding polygons of a set of points; these are polygons that use some of the points as vertices, and contain the rest of them. Removing a mouth from a surrounding polygon ...