Delaunay triangulation

In mathematics, and computational geometry, the Delaunay triangulation or Delone triangularization for a set P of points in the plane is the triangulation DT(P) of P such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid "sliver" triangles. The triangulation was invented by Boris Delaunay in 1934 [1].
In the general ndimensional case it is stated as follows: For a set P of points in the (ndimensional) Euclidean space, the Delaunay triangulation is the triangulation DT(P) of P such that no point in P is inside the circumhypersphere of any simplex in DT(P).
Equivalently, the Delaunay triangulation of a discrete point set P is the geometric dual of the Voronoi tessellation for P.
It is known that the Delaunay triangulation exists and is unique for P, if P is a set of points in general position; that is, no three points are on the same line and no four are on the same circle, for a two dimensional set of points, or no n + 1 points are on the same hyperplane and no n + 2 points are on the same hypersphere, for an ndimensional set of points. An elegant proof of this fact is outlined below. It is worth mentioning, because it reveals connections between the two constructs fundamental for computational and combinatorial geometry.
The problem of finding the Delaunay triangulation of a set of points in ndimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in (n + 1)dimensional space, by giving all points p an extra coordinate equal to p^{2}, taking the bottom side of the convex hull, and mapping back to ndimensional space by deleting the last coordinate. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplexes. A facet not being a simplex implies that n + 2 of the original points lay on the same dhypersphere, and the points were not in general position.
On the other hand, it is easily seen that for the set of three points on the same line there is no Delaunay triangulation (in fact, no triangulation at all). On the other hand, for 4 points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: clearly, the two possible triangulations that split the quadrangle into two triangles satisfy the Delaunay condition.
Generalizations are possible to metrics other than Euclidean. However in these cases the Delaunay triangulation is not guaranteed to exist or be unique.
Contents 
Properties
Let n be the number of points and d the number of dimensions.
 The union of all simplices in the triangulation is the convex hull of the points.
 The Delaunay triangulation contains at most <math>O(n^{\lceil d/2 \rceil})<math> simplices.
 In the plane (d = 2), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face (see Euler characteristic).
 The Delaunay triangulation maximizes the minimum angle. Compared to any other triangulation of the points, the smallest angle in the Delaunay triangulation is at least as large as the smallest angle in any other. The converse is not true: the Delaunay does not necessarily minimize the maximum angle.
Algorithms
All algorithms for computing Delaunay triangulations rely on fast operations for detecting when a point is within a triangle's circumcircle and an efficient data structure for storing triangles and edges. In two dimensions, one way to detect if point D lies in the circumcircle of A, B, C is to evaluate the determinant
 <math>\begin{bmatrix}A_x&A_y&A_x^2 + A_y^2&1\\
B_x&B_y&B_x^2 + B_y^2&1\\ C_x&C_y&C_x^2 + C_y^2&1\\ D_x&D_y&D_x^2 + D_y^2&1\end{bmatrix}<math> This is positive if and only if D lies in the circumcircle.
Incremental
The most straightforward way of computing the Delaunay triangulation is to repeatedly add one vertex at a time, retriangulating the affected parts of the graph. When a vertex is added, a search is done for all triangles' circumcircles containing the vertex. Then, those triangles are removed and that part of the graph retriangulated. Done naively, this results in a running time of O(n^{2}).
A common way to speed up this method is to sort the vertices by the first coordinate, and add them in that order. Then, one only needs to keep track of circumcircles containing points of large of enough first coordinate. The expected running time in two dimensions in this case is O(n^{3/2}) although the worst case continues to be O(n^{2}).
If one inserts the vertices in a random order, the running time is O(n log n) expected.
Another efficient (O(n log n)) incremental algorithm keeps the whole history of the triangulation on the form of a tree. The elements replacing a conflicting element in an insertion are called its children. When a parent is in conflict with a point to be inserted, so are his children. This provides a fast way of getting the list of triangles to remove (which is the slowest part of any incremental insertion algorithm).
Divide and conquer
This algorithm applies to two dimensions only and is due to Guibas and Stolfi. In this algorithm, one recursively draws a line to split the vertices into two sets, then the Delaunay triangulation is computed for each set, 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 worstcase performance.
Sweepline
This is another O(n log n) algorithm for two dimensions. However in most implementations the divide and conquer approach provides better performance.
Applications
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 the fact that we know fast triangulation algorithms. 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 (http://www.cs.cmu.edu/~quake/triangle.html) package.
References
 B. Delaunay, Sur la sphère vide, Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793800, 1934
External links
 http://www.cs.cornell.edu/Info/People/chew/Delaunay.html
 http://goanna.cs.rmit.edu.au/~gl/research/comp_geom/delaunay/delaunay.html
 http://astronomy.swin.edu.au/~pbourke/terrain/triangulate
 http://www2.cs.cmu.edu/~quake/tripaper/triangle0.htmlde:DelaunayTriangulation
fr:Triangulation de Delaunay he:שילוש דלוני pl:Triangulacja Delaunay sl:Delaunayeva triangulacija