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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 simple polygon is monotone with respect to a line L, if any line orthogonal to L intersects the polygon at most twice. 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 ...
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 ...
The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles—polygons that, like triangles, have exactly three convex vertices. As in point set triangulations ...
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 ]
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 ...
Since every tree with more than one vertex has at least two leaves, every triangulated polygon with more than one triangle has at least two ears. Thus, the two ears theorem is equivalent to the fact that every simple polygon has a triangulation. [2] Triangulation algorithms based on this principle have been called ear-clipping algorithms ...
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.