Bowyer–Watson algorithm

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).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.

File:Bowyer-Watson 0.png|First step: insert a node in an enclosing "super"-triangle

File:Bowyer-Watson 1.png|Insert second node

File:Bowyer-Watson 2.png|Insert third node

File:Bowyer-Watson 3.png|Insert fourth node

File:Bowyer-Watson 4.png|Insert fifth (and last) node

File:Bowyer-Watson 6.png|Remove edges with extremes in the super-triangle

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 following pseudocode describes a basic implementation of the Bowyer-Watson algorithm. Its time complexity is $O(n^2)$. 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 $O(n\; \backslash log\; n)$. Pre-computing the circumcircles can save time at the expense of additional memory usage. And if the points are uniformly distributed, sorting them along a space filling Hilbert curve prior to insertion can also speed point location.Liu, Yuanxin, and Jack Snoeyink. "A comparison of five implementations of 3D Delaunay tessellation." Combinatorial and Computational Geometry 52 (2005): 439-458.

function BowyerWatson (pointList)

// pointList is a set of coordinates defining the points to be triangulated

triangulation := empty triangle mesh data structure

add super-triangle to triangulation // must be large enough to completely contain all the points in pointList

for each point in pointList do // add all the points one at a time to the triangulation

badTriangles := empty set

for each triangle in triangulation do // first find all the triangles that are no longer valid due to the insertion

if point is inside circumcircle of triangle

add triangle to badTriangles

polygon := empty set

for each triangle in badTriangles do // find the boundary of the polygonal hole

for each edge in triangle do

if edge is not shared by any other triangles in badTriangles

add edge to polygon

for each triangle in badTriangles do // remove them from the data structure

remove triangle from triangulation

for each edge in polygon do // re-triangulate the polygonal hole

newTri := form a triangle from edge to point

add newTri to triangulation

for each triangle in triangulation // done inserting points, now clean up

if triangle contains a vertex from original super-triangle

remove triangle from triangulation

return triangulation

{{reflist}}

• {{Cite journal | last1 = Bowyer | first1 = Adrian |author1-link=Adrian Bowyer| title = Computing Dirichlet tessellations | doi = 10.1093/comjnl/24.2.162 | journal = Comput. J. | volume = 24 | issue = 2 | pages = 162–166 | year = 1981 | doi-access = free }}
• {{Cite journal | last1 = Watson | first1 = David F. | title = Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes | doi = 10.1093/comjnl/24.2.167 | journal = Comput. J. | volume = 24 | issue = 2 | pages = 167–172 | year = 1981 | doi-access = }}
• [http://paulbourke.net/papers/triangulate/ Efficient Triangulation Algorithm Suitable for Terrain Modelling] generic explanations with source code examples in several languages.

• [https://github.com/jmespadero/pyDelaunay2D pyDelaunay2D] : A didactic Python implementation of Bowyer–Watson algorithm.
• [https://github.com/Bl4ckb0ne/delaunay-triangulation Bl4ckb0ne/delaunay-triangulation] : C++ implementation of Bowyer–Watson algorithm.

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Category:Geometric algorithms

Category:Triangulation (geometry)