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The time complexity of triangulation of an n-vertex polygon with holes has an Ω(n log n) lower bound, in algebraic computation tree models of computation. [1] It is possible to compute the number of distinct triangulations of a simple polygon in polynomial time using dynamic programming , and (based on this counting algorithm) to generate ...
An ear of a polygon is defined as a triangle formed by three consecutive vertices ,, of the polygon, such that its edge lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. [1]
[4] [5] The run-time complexity of this problem crucially depends on whether the raw polygon is allowed to have holes. If the raw polygon is hole-free , then an optimal partition can be found in time O ( n 4 ) {\displaystyle O(n^{4})} , where n is the number of vertices of the polygon.
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 ().
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.
In computational geometry, a constrained Delaunay triangulation is a generalization of the Delaunay triangulation that forces certain required segments into the triangulation as edges, [1] [2] unlike the Delaunay triangulation itself which is based purely on the position of a given set of vertices without regard to how they should be connected by edges.
In computational geometry and computer science, the minimum-weight triangulation problem is the problem of finding a triangulation of minimal total edge length. [1] That is, an input polygon or the convex hull of an input point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the ...
The Greedy Triangulation is a method to compute a polygon triangulation or a Point set triangulation using a greedy schema, which adds edges one by one to the solution in strict increasing order by length, with the condition that an edge cannot cut a previously inserted edge. [1] [2]