When a vertex v is added, we split in three the triangle that contains v , then we apply the flip algorithm. Done naively, this will take O n time: we search through all the triangles to find the one that contains v , then we potentially flip away every triangle. Then the overall runtime is O n 2.
If we insert vertices in random order, it turns out by a somewhat intricate proof that each insertion will flip, on average, only O 1 triangles -- although sometimes it will flip many more [4].
This still leaves the point location time to improve. We can store the history of the splits and flips performed: each triangle stores a pointer to the two or three triangles that replaced it.
To find the triangle that contains v , we start at a root triangle, and follow the pointer that points to a triangle that contains v , until we find a triangle that has not yet been replaced. On average, this will also take O log n time.
Over all vertices, then, this takes O n log n time [3]. While the technique extends to higher dimension as proved by Edelsbrunner and Shah [5] , the runtime can be exponential in the dimension even if the final Delaunay triangulation is small. A divide and conquer algorithm for triangulations in two dimensions is due to Lee and Schachter which was improved by Guibas and Stolfi [6] and later by Dwyer.
In this algorithm, one recursively draws a line to split the vertices into two sets. The Delaunay triangulation is computed for each set, and then the two sets are merged along the splitting line. Using some clever tricks, the merge operation can be done in time O n , so the total running time is O n log n. For certain types of point sets, such as a uniform random distribution, by intelligently picking the splitting lines the expected time can be reduced to O n log log n while still maintaining worst-case performance.
A divide and conquer paradigm to performing a triangulation in d -dimensions is presented in "DeWall: A fast divide and conquer Delaunay triangulation algorithm in E d " by P. Cignoni, C. Montani, R. Fortune's Algorithm uses a sweepline technique to achieve O n log n runtime in the planar case. In some experimental evaluations, [ citation needed ] the sweepline is the fastest in practice. The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points, and this can be exploited to compute it efficiently.
For modeling terrain or other objects given a set of sample points, the Delaunay triangulation gives a nice set of triangles to use as polygons in the model. In particular, the Delaunay triangulation avoids narrow triangles as they have large circumcircles compared to their area. Delaunay triangulations are often used to build meshes for the finite element method, because of the angle guarantee and because fast triangulation algorithms have been developed.
Typically, the domain to be meshed is specified as a coarse simplicial complex ; for the mesh to be numerically stable, it must be refined, for instance by using Ruppert's algorithm. This has been implemented by Jonathan Shewchuk in the freely available Triangle package. Delaunay triangulation From wiki.
Jump to: navigation , search. The Delaunay triangulation with all the circumcircles and their centers in red. Flipping the common edge produces a Delaunay triangulation for the four points.
Retrieved Computational Geometry: Algorithms and Applications. ISBN Knuth; M. Sharir Algorithmica 7 : — Algorithmica 15 : — Montani; R. Scopigno Okabe, A. New York: Wiley, Preparata, F.
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