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A polygon ear. One way to triangulate a simple polygon is based on the two ears theorem, as the fact that any simple polygon with at least 4 vertices without holes has at least two "ears", which are triangles with two sides being the edges of the polygon and the third one completely inside it. [5]
For a hole-free polygon with vertices, a triangulation can be calculated in time (). For a polygon with holes , there is a lower bound of Ω ( n log n ) {\displaystyle \Omega (n\log n)} . A related problem is partitioning to triangles with a minimal total edge length, also called minimum-weight triangulation .
However, this polygon also has other ears that are not evident in this triangulation. In geometry , the two ears theorem states that every simple polygon with more than three vertices has at least two ears , vertices that can be removed from the polygon without introducing any crossings.
In geometry, a polygon with holes is an area-connected planar polygon with one external boundary and one or more interior boundaries (holes). [1] Polygons with holes can be dissected into multiple polygons by adding new edges, so they are not frequently needed. An ordinary polygon can be called simply-connected, while a polygon-with-holes is ...
Polygon triangulations may be found in linear time and form the basis of several important geometric algorithms, including a simple approximate solution to the art gallery problem. The constrained Delaunay triangulation is an adaptation of the Delaunay triangulation from point sets to polygons or, more generally, to planar straight-line graphs.
The following pseudocode describes a basic implementation of the Bowyer-Watson algorithm. Its time complexity is ().Efficiency can be improved in a number of ways. For example, the triangle connectivity can be used to locate the triangles which contain the new point in their circumcircle, without having to check all of the triangles - by doing so we can decrease time complexity to ().
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]
Three possible triangulations of the same polygon. The central triangulation has less weight (sum of perimeters). In computational geometry and computer science, the minimum-weight triangulation problem is the problem of finding a triangulation of minimal total edge length. [1]