mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

DT(P) such that no point in P is inside the

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of any

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 Boris Nikolayevich Delaunay or Delone (russian: Бори́с Никола́евич Делоне́; 15 March 1890 – 17 July 1980) was a Soviet and Russian mathematician, mountain climber, and the father of physicist, Nikolai Borisovich Delone. ...

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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

other than

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...

. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique.

Relationship with the Voronoi diagram

The Delaunay

of a

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

point set P in

general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...

corresponds to the

dual graph In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop ...

of the

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...

for P. The

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

s 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: * Three or more

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

points, where the circumcircles are of infinite

radii In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

. * Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical. *Edges of the Voronoi diagram going to infinity are not defined by this relation in case of a finite set P. If the Delaunay

is calculated using the

Bowyer–Watson algorithm 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 also used to obtain a Voronoi diagram of the points, which is ...

then the circumcenters of triangles having a common vertex with the "super" triangle should be ignored. Edges going to infinity start from a circumcenter and they are perpendicular to the common edge between the kept and ignored triangle.

''d''-dimensional Delaunay

For a set P of points in the (''d''-dimensional)

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, a Delaunay triangulation is a

DT(P) such that no point in P is inside the circum-hypersphere of any ''d''-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

in DT(P). It is known that there exists a unique Delaunay triangulation for P if P is a set of points in ''

''; 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

can be converted to the problem of finding the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

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 In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

. Nonsimplicial facets only occur when ''d'' + 2 of the original points lie on the same ''d''-

hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...

, i.e., the points are not in general position.

Properties

Example steps in Delauney triangularization

Delaunay triangulation does not minimize edge length

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 ''O''(''n''^{⌈''d'' / 2⌉}) simplices. * In the plane (''d'' = 2), if there are ''b'' vertices on the convex hull, then any triangulation of the points has at most 2''n'' − 2 − ''b'' triangles, plus one exterior face (see

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

). * If points are distributed according to a

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

in the plane with constant intensity, then each vertex has on average six surrounding triangles. More generally for the same process in ''d'' dimensions the average number of neighbors is a constant depending only on ''d''. * In the plane, 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. However, the Delaunay triangulation does not necessarily minimize the maximum angle. The Delaunay triangulation also does not necessarily minimize the length of the edges. * A circle circumscribing any Delaunay triangle does not contain any other input points in its interior. * If a circle passing through two of the input points doesn't contain any other input points in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points. * Each triangle of the Delaunay triangulation of a set of points in ''d''-dimensional spaces corresponds to a facet of

of the projection of the points onto a (''d'' + 1)-dimensional

paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane ...

, and vice versa. * The closest neighbor ''b'' to any point ''p'' is on an edge ''bp'' in the Delaunay triangulation since the

nearest neighbor graph The nearest neighbor graph (NNG) is a directed graph defined for a set of points in a metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from ''p'' to ''q'' whenever ''q'' is a nea ...

is a subgraph of the Delaunay triangulation. * The Delaunay triangulation is a

geometric spanner A geometric spanner or a -spanner graph or a -spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a -path between any pair of vertices for a fixed parameter . A -path is defined as a path ...

: In the plane (''d'' = 2), the shortest path between two vertices, along Delaunay edges, is known to be no longer than 1.998 times the Euclidean distance between them.

Visual Delaunay definition: Flipping

From the above properties an important feature arises: Looking at two triangles ABD and BCD with the common edge BD (see figures), if the sum of the angles α and γ is less than or equal to 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: File:Delaunay geometry.png, This triangulation does not meet the Delaunay condition (the sum of α and γ is bigger than 180°). File:Point inside circle - Delaunay condition broken.svg, This pair of triangles does not meet the Delaunay condition (there is a point within the interior of the circumcircle). File:Edge Flip - Delaunay condition ok.svg, ''Flipping'' the common edge produces a valid Delaunay triangulation for the four points. This operation is called a ''flip'', and can be generalised to three and higher dimensions.

Algorithms

Point inside circle - Delaunay condition broken - Labelled

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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

: :

= & \begin A_x - D_x & A_y - D_y & (A_x^2 - D_x^2) + (A_y^2 - D_y^2) \\ B_x - D_x & B_y - D_y & (B_x^2 - D_x^2) + (B_y^2 - D_y^2) \\ C_x - D_x & C_y - D_y & (C_x^2 - D_x^2) + (C_y^2 - D_y^2) \end > 0 \end

When ''A'', ''B'' and ''C'' are sorted in a

counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

order, this determinant is positive only if ''D'' lies inside the circumcircle.

Flip algorithms

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(''n''²) 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 In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are spec ...

: this graph is connected for two-dimensional sets of points, but may be disconnected in higher dimensions.

Incremental

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(''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. 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. 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

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.Blelloch, Guy; Gu, Yan; Shun, Julian; and Sun, Yihan
Parallelism in Randomized Incremental Algorithms
. SPAA 2016. doi:10.1145/2935764.2935766. proposed another version of incremental algorithm based on rip-and-tent, which is practical and highly parallelized with polylogarithmic

span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...

Divide and conquer

divide and conquer algorithm In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved dire ...

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 E^''d''" by P. Cignoni, C. Montani, R. Scopigno. The divide and conquer algorithm has been shown to be the fastest DT generation technique sequentially.

Sweephull

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.

Applications

The

Euclidean minimum spanning tree A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments ...

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 Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin word ...

or other objects given a

point cloud Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Poin ...

, 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 In computer graphics, a triangulated irregular network (TIN) is a representation of a continuous surface consisting entirely of triangular facets (a triangle mesh), used mainly as Discrete Global Grid in primary elevation modeling. The vertic ...

. Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the Delaunay tessellation field estimator (DTFE). Delaunay Triangulation (100 Points)

Delaunay triangulations are often used to generate meshes for space-discretised solvers such as the

finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...

and the

finite volume method The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...

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 In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...

; for the mesh to be numerically stable, it must be refined, for instance by using

Ruppert's algorithm In mesh generation, Delaunay refinement are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulat ...

. The increasing popularity of

and

boundary element method The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, el ...

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 In computational geometry, a constrained Delaunay triangulation is a generalization of the Delaunay triangulation that forces certain required segments into the triangulation as edges, unlike the Delaunay triangulation itself which is based purely ...

has found applications in path planning in automated driving and topographic surveying.

References

External links

*
Delaunay triangulation
. Wolfram MathWorld. Retrieved April 2010.

Software

* Delaunay triangulation in CGAL, the Computational Geometry Algorithms Library: ** Mariette Yvinec
2D Triangulation
Retrieved April 2010. ** Pion, Sylvain; Teillaud, Monique
3D Triangulations
Retrieved April 2010. ** Hornus, Samuel; Devillers, Olivier; Jamin, Clément
dD Triangulations
** Hert, Susan; Seel, Michael
dD Convex Hulls and Delaunay Triangulations
Retrieved April 2010. *
Poly2Tri: Incremental constrained Delaunay triangulation
Open source C++ implementation. Retrieved April 2019. *
Divide & Conquer Delaunay triangulation construction
. Open source C99 implementation. Retrieved April 2019. *
CDT: Constrained Delaunay Triangulation in C++
. Open source C++ implementation. Retrieved August 2022. {{DEFAULTSORT:Delaunay Triangulation Triangulation (geometry) Geometric algorithms