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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 ...
Minimum-weight triangulation is a triangulation in which the goal is to minimize the total edge length. A point-set triangulation is a polygon triangulation of the convex hull of a set of points. A Delaunay triangulation is another way to create a triangulation based on a set of points.
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 .
The Delaunay triangulation of a set of points in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of . Triangulations have a number of applications, and there is an interest to find the "good" triangulations of a given point set under some criteria as, for instance minimum-weight triangulations .
Similarly, the minimum-weight triangulation of planar point sets is NP-hard for arbitrary point sets, [4] but solvable in polynomial time by dynamic programming for points in convex position. [ 5 ] The Erdős–Szekeres theorem guarantees that every set of n {\displaystyle n} points in general position (no three in a line) in two or more ...
The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points, [22] and this can be exploited to compute it efficiently. For modelling terrain or other objects given a point cloud , the Delaunay triangulation gives a nice set of triangles to use as polygons in the model.
Shares of the Indiana-based drugmaker rose 2.6% to $905.32 after Lilly's announcement on Tuesday, while stocks of telehealth companies that offer compounded versions of weight-loss drugs fell in ...
Frequently used and studied point set triangulations include the Delaunay triangulation (for points in general position, the set of simplices that are circumscribed by an open ball that contains no input points) and the minimum-weight triangulation (the point set triangulation minimizing the sum of the edge lengths).