In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their

convex hull In geometry, the convex hull, 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, ...

into triangles whose

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

s do not contain any of the points; that is, each circumcircle has its generating points on its circumference, but all other points in the set are outside of it. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles. The triangulation is named after

Boris Delaunay Boris Nikolayevich Delaunay or Delone (; 15 March 1890 – 17 July 1980) was a Soviet and Russian mathematician, mountain climber, and the father of physicist, Nikolai Borisovich Delone. He is best known for the Delaunay triangulation. Biograph ...

for his work on it from 1934. If the points all lie on a straight line, the notion of triangulation becomes degenerate and there is no Delaunay triangulation. 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: Measuring * 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 ...

other than

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

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

Relationship with the Voronoi diagram

The Delaunay

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 m ...

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, ...

point set in general position corresponds to the

dual graph In the mathematics, mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex (graph theory), vertex for each face (graph theory), face of . The dual graph has an edge (graph theory), edge for each p ...

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. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

for . The

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

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 Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

points, where the circumcircles are of infinite

radii In classical geometry, a radius (: radii or radiuses) 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 radius of a regular polygon is the line segment or ...

. * Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical. In this case the Voronoi diagram contains vertices of degree four or greater and its dual graph contains polygonal faces with four or more sides. The various triangulations of these faces complete the various possible Delaunay triangulations. *Edges of the Voronoi diagram going to infinity are not defined by this relation in case of a finite set . 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 i ...

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 of points in the (-dimensional)

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

, a Delaunay triangulation is a

such that no point in is inside the circum-hypersphere of any -

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 . It is known that there exists a unique Delaunay triangulation for if is a set of points 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 a ...

''; that is, the affine hull of is -dimensional and no set of points in lie on the boundary of a ball whose interior does not intersect . The problem of finding the Delaunay triangulation of a set of points in -dimensional

can be converted to the problem of finding the

of a set of points in ()-dimensional space. This may be done by giving each point an extra coordinate equal to , 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 -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 of the original points lie on the same -

hypersphere In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point ...

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

Properties

Let be the number of points and the number of dimensions. * The union of all simplices in the triangulation is the convex hull of the points. * The Delaunay triangulation contains simplices. * In the plane (), if there are vertices on the convex hull, then any triangulation of the points has at most 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's ...

). * If points are distributed according to a

Poisson process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...

in the plane with constant intensity, then each vertex has on average six surrounding triangles. More generally for the same process in dimensions the average number of neighbors is a constant depending only on . * 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 -dimensional spaces corresponds to a facet of

of the projection of the points onto a ()-dimensional

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

, and vice versa. * The closest neighbor to any point is on an edge 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 n ...

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 (), 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 with the common edge (see figures), if the sum of the angles , 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 for the common edge produces two triangles that do meet the Delaunay condition: File:Delaunay geometry.png, This triangulation does not meet the Delaunay condition (the sum of and 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 lies in the circumcircle of is to evaluate the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

: :

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

When are sorted in a

counterclockwise Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...

order, this determinant is positive only if 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 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.

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 is added, we split in three the triangle that contains , then we apply the flip algorithm. Done naïvely, this will take time: we search through all the triangles to find the one that contains , then we potentially flip away every triangle. Then the overall runtime is . If we insert vertices in random order, it turns out (by a somewhat intricate proof) that each insertion will flip, on average, only 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 , we start at a root triangle, and follow the pointer that points to a triangle that contains , until we find a triangle that has not yet been replaced. On average, this will also take time. Over all vertices, then, this takes 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 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.

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 dir ...

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 , so the total running time is . 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 while still maintaining worst-case performance. A divide and conquer paradigm to performing a triangulation in 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 (), alternatively relief or topographical relief, is the dimension and shape of a given surface of land. In physical geography, terrain is the lay of the land. This is usually expressed in terms of the elevation, slope, and orientati ...

or other objects given a

point cloud A point cloud is a discrete set of data Point (geometry), points in space. The points may represent a 3D shape or object. Each point Position (geometry), position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than ...

, 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 verti ...

. 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 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 tran ...

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 divergen ...

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 structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

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

Ruppert's algorithm In mesh generation, Delaunay refinements are algorithms for mesh generation based on the principle of adding Steiner point (computational geometry), Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangul ...

. 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, ele ...

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 The stretched grid method (SGM) is a Numerical analysis, numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior. In particular, meteorologists use the ...

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 Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used ...

in automated driving and topographic surveying.

References

External links

* Henry, Ian (July 11, 2022).
Visualizing Delaunay Triangulation
. Blog post detailing algorithms for Delaunay triangulation. * Delaunay triangulation in

CGAL The Computational Geometry Algorithms Library (CGAL) is an open source software library of computational geometry algorithms. While primarily written in C++, Scilab bindings and bindings generated with SWIG (supporting Python and Java for now) a ...

, 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