Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation ; they tend to avoid skinny triangles.

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.

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.

Delaunay triangulations are often used to build 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.

In some contexts, e. g., when discussing Voronoi tessellations and Delaunay triangulations in the plane, a stricter definition is used: a set of points in the plane is then said to be in general position only if no three of them lie on the same straight line and no four lie on the same circle.

** Voronoi diagrams and Delaunay triangulations

The proof is based on two properties of minimum spanning trees and Delaunay triangulations:

#( a property of Delaunay triangulations ): If there is a circle with two of the input points on its boundary which contains no other input points, the line between those two points is an edge of every Delaunay triangulation.

There are special triangulations like the Delaunay triangulation which is the geometric dual of the Voronoi diagram.

Frequently used and studied point set triangulations include the Delaunay triangulation ( for points in general position, the set of triangles defined by three of the input points and not containing a fourth input point ), and the minimum-weight triangulation ( the point set triangulation minimizing the sum of the edge lengths ).

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, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

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 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.

If the input coordinates are integers and can be used as array indices, faster algorithms are possible: the Delaunay triangulation can be constructed by a randomized algorithm in O ( n log log n ) expected time.

Additionally, since the Delaunay triangulation is a planar graph, its minimum spanning tree can be found in linear time by a variant of Borůvka's algorithm that removes all but the cheapest edge between each pair of components after each stage of the algorithm.

All edges of an EMST are edges of a relative neighborhood graph, which in turn are edges of a Gabriel graph, which are edges in a Delaunay triangulation of the points, as can be proven via the equivalent contrapositive statement: every edge not in a Delaunay triangulation is also not in any EMST.

Joscelin accompanies her to the marquist's, but before the marque can be finished, they are interrupted by a sailor, bearing a message from Admiral Quintilius Rousse to Delaunay ; he knows that Delaunay's house is being watched, and seeks to give the message to Phèdre instead.

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

It is difficult to apply to painters such as Jean Metzinger, Albert Gleizes, Robert Delaunay and Henri Le Fauconnier, whose fundamental differences from traditional Cubism compelled Kahnweiler to question their right to be called Cubists at all.

Metzinger had already written in 1910 of ' mobile perspective ', as an interpretation of what would soon be dubbed " Cubism " with respect to Picasso, Braque, Delaunay and Le Fauconnier ( Metzinger, Note sur la peinture, Pan, Paris, Oct-Nov 1910 ).

Her dissatisfaction with her position had become so evident that the duchess, afraid of losing her services, arranged the marriage to give Mlle Delaunay rank sufficient to allow of her promotion to be on an equality with the ladies of the court.

Macke's meeting with Robert Delaunay in Paris in 1912 was to be a sort of revelation for him.

However, the word " cube " was used in 1906 by another critic, Louis Chassevent, with reference not to Picasso or Braque but rather to Metzinger and Delaunay: " M. Metzinger is a mosaicist like M. Signac but he brings more precision to the cutting of his cubes of color which appear to have been made mechanically [...]".

The spelling Delone is a straightforward transliteration from Cyrillic he often used in recent publications, while Delaunay is French language version he used in the early French and German publications.

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT ( P ) such that no point in P is inside the circumcircle of any triangle in DT ( P ).

For a set of points on the same line there is no Delaunay triangulation ( the notion of triangulation is degenerate for this case ).

The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi tessellation for P. Special cases include the existence of three points on a line and four points on circle.

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 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, there exists no k-flat containing k + 2 points nor a k-sphere containing k + 3 points, for 1 ≤ k ≤ d − 1 ( e. g., for a set of points in < big > ℝ </ big >< sup > 3 </ sup >; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere ).

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.

* 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 of them in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points.

* The each triangle of the Delaunay triangulation of a set of points in d-dimensional spaces corresponds to a facet of convex hull of the projection of the points onto a ( d + 1 )- dimensional paraboloid, and vice versa.

Image: Delaunay_before_flip. png | This triangulation does not meet the Delaunay condition ( the circumcircles contain more than three points ).

Image: Delaunay_after_flip. png | Flipping the common edge produces a Delaunay triangulation for the four points.

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 < sup > d </ sup >" by P. Cignoni, C. Montani, R. Scopigno .< ref >

Adams's finding provoked a sharp astronomical controversy that lasted some years, but the correctness of his result, agreed by other mathematical astronomers including C E Delaunay, was eventually accepted.

A part of the answer was suggested independently in the 1860s by Delaunay and by William Ferrel: tidal retardation of the Earth's rotation rate was lengthening the unit of time and causing a lunar acceleration that was only apparent.

Cubism is an early-20th-century avant-garde art movement pioneered by Pablo Picasso and Georges Braque, and later joined by Juan Gris, Jean Metzinger, Albert Gleizes, Robert Delaunay, Henri Le Fauconnier, and Fernand Léger, that revolutionized European painting and sculpture, and inspired related movements in music, literature and architecture.

The first organized group exhibition by Cubists took place at the Salon des Indépendants in Paris during the spring of 1911 in a room called ‘ Salle 41 ’; it included works by Jean Metzinger, Albert Gleizes, Fernand Léger, Robert Delaunay and Henri Le Fauconnier, yet no works by Picasso and Braque were exhibited.

This showing by Metzinger, Gleizes, Delaunay, le Fauconnier and Léger brought Cubism to the attention of the general public for the first time.

Other Cubists, by contrast, especially František Kupka, and those considered Orphists by Apollinaire ( Delaunay, Léger, Picabia and Duchamp ), accepted abstraction by removing visible subject matter entirely.

From 1912 Delaunay painted a series of paintings entitled Simultaneous Windows, followed by a series entitled Formes Circulaires, in which he combined planar structures with bright prismatic hues ; based on the optical characteristics of juxtaposed colors his departure from reality in the depiction of imagery was quasi-complete.

Geneviève Bizet, painted in 1878 by Jules-Élie Delaunay.

The death of the Nymph Hesperides | Hesperia by Elie Delaunay.

Ixion by Jules-Elie Delaunay

S. A. des Automobiles Delaunay-Belleville was formed in 1903 by Louis Delaunay and Marius Barbarou.

International artists are few in the collection, but there are works by Robert Delaunay, Yves Tanguy, Man Ray, Jacques Lipchitz, Lucio Fontana, Yves Klein, Max Ernst, Richard Serra, Bruce Nauman, Donald Judd, Damien Hirst, Julian Schnabel, Joseph Beuys, Nam June Paik, Wolf Vostell, Gabriel Orozco, Clyfford Still, cubist still lifes by Georges Braque and a large work by Francis Bacon.