In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934.
For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.
By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique.
The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. In the 2D case, the Voronoi vertices are connected via edges, that can be derived from adjacency-relationships of the Delaunay triangles: If two triangles share an edge in the Delaunay triangulation, their circumcenters are to be connected with an edge in the Voronoi tesselation.
Special cases where this relationship does not hold, or is ambiguous, include cases like:
For a set P of points in the (d-dimensional) Euclidean space, a Delaunay triangulation is a triangulation DT(P) such that no point in P is inside the circum-hypersphere of any d-simplex in DT(P). It is known that there exists a unique Delaunay triangulation for P if P is a set of points in general position; that is, the affine hull of P is d-dimensional and no set of d + 2 points in P lie on the boundary of a ball whose interior does not intersect P.
The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in (d + 1)-dimensional space. This may be done by giving each point p an extra coordinate equal to |p|2, thus turning it into a hyper-paraboloid (this is termed "lifting"); taking the bottom side of the convex hull (as the top end-cap faces upwards away from the origin, and must be discarded); and mapping back to d-dimensional space by deleting the last coordinate. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplices. Nonsimplicial facets only occur when d + 2 of the original points lie on the same d-hypersphere, i.e., the points are not in general position.
Let n be the number of points and d the number of dimensions.
From the above properties an important feature arises: Looking at two triangles △ABD, △BCD with the common edge BD (see figures), if the sum of the angles α + γ ≤ 180°, the triangles meet the Delaunay condition.
This is an important property because it allows the use of a flipping technique. If two triangles do not meet the Delaunay condition, switching the common edge BD for the common edge AC produces two triangles that do meet the Delaunay condition:
This operation is called a flip, and can be generalised to three and higher dimensions.
Many 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:
When A, B, C are sorted in a counterclockwise order, this determinant is positive only if D lies inside the circumcircle.
As mentioned above, if a triangle is non-Delaunay, we can flip one of its edges. This leads to a straightforward algorithm: construct any triangulation of the points, and then flip edges until no triangle is non-Delaunay. Unfortunately, this can take Ω(n2) edge flips. While this algorithm can be generalised to three and higher dimensions, its convergence is not guaranteed in these cases, as it is conditioned to the connectedness of the underlying flip graph: this graph is connected for two-dimensional sets of points, but may be disconnected in higher dimensions.
The most straightforward way of efficiently computing the Delaunay triangulation is to repeatedly add one vertex at a time, retriangulating the affected parts of the graph. When a vertex v is added, we split in three the triangle that contains v, then we apply the flip algorithm. Done naïvely, 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(n2).
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. 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 containsv, we start at a root triangle, and follow the pointer that points to a triangle that containsv, 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. While the technique extends to higher dimension (as proved by Edelsbrunner and Shah), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.
The Bowyer–Watson algorithm provides another approach for incremental construction. It gives an alternative to edge flipping for computing the Delaunay triangles containing a newly inserted vertex.
Unfortunately the flipping-based algorithms are generally hard to parallelize, since adding some certain point (e.g. the center point of a wagon wheel) can lead to up to O(n) consecutive flips. Blelloch et al. proposed another version of incremental algorithm based on rip-and-tent, which is practical and highly parallelized with polylogarithmic span.
A divide and conquer algorithm for triangulations in two dimensions was developed by Lee and Schachter and improved by Guibas and Stolfi 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 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 Ed" by P. Cignoni, C. Montani, R. Scopigno.
The divide and conquer algorithm has been shown to be the fastest DT generation technique sequentially.
Sweephull is a hybrid technique for 2D Delaunay triangulation that uses a radially propagating sweep-hull, and a flipping algorithm. The sweep-hull is created sequentially by iterating a radially-sorted set of 2D points, and connecting triangles to the visible part of the convex hull, which gives a non-overlapping triangulation. One can build a convex hull in this manner so long as the order of points guarantees no point would fall within the triangle. But, radially sorting should minimize flipping by being highly Delaunay to start. This is then paired with a final iterative triangle flipping step.
See also: Voronoi diagram § 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 modelling terrain or other objects given a point cloud, 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). See triangulated irregular network.
Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the Delaunay tessellation field estimator (DTFE).
Delaunay triangulations are often used to generate meshes for space-discretised solvers such as the finite element method and the finite volume method of physics simulation, 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.
The increasing popularity of finite element method and boundary element method techniques increases the incentive to improve automatic meshing algorithms. However, all of these algorithms can create distorted and even unusable grid elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. For example, smoothing (also referred to as mesh refinement) is one such method, which repositions nodes to minimize element distortion. The stretched grid method allows the generation of pseudo-regular meshes that meet the Delaunay criteria easily and quickly in a one-step solution.
Constrained Delaunay triangulation has found applications in path planning in automated driving and topographic surveying. 
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