Bowyer–Watson algorithm: Difference between revisions
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In [[computational geometry]], the '''Bowyer–Watson algorithm''' is a method for computing the [[Delaunay triangulation]] of a finite set of points in any number of [[dimension]]s. The algorithm can be used to obtain a [[Voronoi diagram]] of the points, which is the [[dual graph]] of the Delaunay triangulation. |
In [[computational geometry]], the '''Bowyer–Watson algorithm''' is a method for computing the [[Delaunay triangulation]] of a finite set of points in any number of [[dimension]]s. The algorithm can be used to obtain a [[Voronoi diagram]] of the points, which is the [[dual graph]] of the Delaunay triangulation. |
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The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a [[star-shaped polygon]]al hole which is then re-triangulated using the new point. |
The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a [[star-shaped polygon]]al hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take ''O(N log N)'' operations to triangulate N points, although special degenerate cases exist where this goes up to ''O(N<sup>2</sup>)''.<ref>Rebay, S. ''Efficient Unstructured Mesh Generation by Means of Delaunay Triangulation and Bowyer-Watson Algorithm''. Journal of Computational Physics Volume 106 Issue 1, May 1993, p. 127.</ref> |
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The algorithm is sometimes known just as the '''Bowyer Algorithm''' or the '''Watson Algorithm'''. [[Adrian Bowyer]] and David Watson devised it independently of each other at the same time, and each published a paper on it in the same issue of ''[[The Computer Journal]]'' (see below). |
The algorithm is sometimes known just as the '''Bowyer Algorithm''' or the '''Watson Algorithm'''. [[Adrian Bowyer]] and David Watson devised it independently of each other at the same time, and each published a paper on it in the same issue of ''[[The Computer Journal]]'' (see below). |
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Revision as of 18:52, 5 August 2014
In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation.
The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take O(N log N) operations to triangulate N points, although special degenerate cases exist where this goes up to O(N2).[1]
The algorithm is sometimes known just as the Bowyer Algorithm or the Watson Algorithm. Adrian Bowyer and David Watson devised it independently of each other at the same time, and each published a paper on it in the same issue of The Computer Journal (see below).
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First step: insert a node in an enclosing "super"-triangle -
Insert second node -
Insert third node -
Insert fourth node -
Insert fifth (and last) node -
Remove super-triangle edges
See also
References
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|doi=10.1093/comjnl/24.2.167instead. - Efficient Triangulation Algorithm Suitable for Terrain Modelling generic explanations with source code examples in several languages.
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- ^ Rebay, S. Efficient Unstructured Mesh Generation by Means of Delaunay Triangulation and Bowyer-Watson Algorithm. Journal of Computational Physics Volume 106 Issue 1, May 1993, p. 127.