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The closest neighbor b to any point p is on an edge bp in the Delaunay triangulation since the nearest neighbor graph is a subgraph of the Delaunay triangulation. The Delaunay triangulation is a geometric spanner : In the plane ( d = 2 ), the shortest path between two vertices, along Delaunay edges, is known to be no longer than 1.998 times the ...
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 ().
The input to the constrained Delaunay triangulation problem is a planar straight-line graph, a set of points and non-crossing line segments in the plane.The constrained Delaunay triangulation of this input is a triangulation of its convex hull, including all of the input segments as edges, and using only the vertices of the input.
The Gabriel graph is a subgraph of the Delaunay triangulation. It can be found in linear time if the Delaunay triangulation is given. [4] The Gabriel graph contains, as subgraphs, the Euclidean minimum spanning tree, the relative neighborhood graph, and the nearest neighbor graph. It is an instance of a beta-skeleton.
Diagram of the Delaunay Triangulation of 100 random points. Plotted using Mathematica. Code (and points used) given below. Date: 13 April 2007: Source: Own drawing, Inkscape 0.45: Author: Inductiveload: Permission (Reusing this file)
Diagram of the Delaunay Triangulation of 25 random points. Plotted using Mathematica. Code (and points used) given below. Date: 13 April 2007: Source: Own drawing, Inkscape 0.45: Author: Inductiveload: Permission (Reusing this file)
Example of Urquhart graph: the (thin cyan) longest edges are removed from each Delaunay triangle.. In computational geometry, the Urquhart graph of a set of points in the plane, named after Roderick B. Urquhart, is obtained by removing the longest edge from each triangle in the Delaunay triangulation.
The algorithm begins with a constrained Delaunay triangulation of the input vertices. At each step, the circumcenter of a poor-quality triangle is inserted into the triangulation with one exception: If the circumcenter lies on the opposite side of an input segment as the poor quality triangle, the midpoint of the segment is inserted. Moreover ...