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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 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 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 .
Polygon triangulation. In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) P into a set of triangles, [1] i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is P. Triangulations may be viewed as special cases of planar straight-line graphs.
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.
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.
In the Pitteway triangulation, each edge pq either belongs to the convex hull or crosses the edge of the Voronoi diagram that separates the cells containing p and q. In some references this property is used to define a Pitteway triangulation, as a Delaunay triangulation in which all internal Delaunay edges cross their dual Voronoi edges.
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 ...