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| [[Image:Delaunay_circumcircles_vectorial.svg|right|thumb|280px|A Delaunay triangulation in the plane with circumcircles shown]]
| | Some study database organizations quite often invite members to participate in surveys in substitution for gift vouchers, entry in a draw or some other kind of reward,... <br><br>Can you really receive money to take short online surveys? If you"ve been considering approaches to earn money from home, you will undoubtedly have encounter paid internet surveys. The initial question most people desire to ask is if they really can receive money to take short internet surveys. The clear answer is just a definite "yes." <br><br>Some survey database organizations very often ask members to be involved in studies in substitution for present vouchers, entry in a draw or some other kind of reward, but members will also be provided regular chances to receive money cash to have a small online survey. There are people who try to make money completing surveys but give up as the first few surveys they"re presented are for prize draws as opposed to for cash. <br><br>These are the folks who go online complaining that paid online surveys are all scams. It is a pity that, because a half-hearted attempt was only made by them at taking part in internet surveys, these folks can spoil the opportunity for other individuals who are looking for ways to make money from your home. It"s part of human nature that individuals with a are all too wanting to [http://Search.huffingtonpost.com/search?q=discuss&s_it=header_form_v1 discuss] their views while happy rabbits, just get on with what they"re doing. <br><br>If a survey database company is joined by you, don"t lose heart if you are presented studies to complete in substitution for entry in a draw or some other honor. You are not required to complete the survey; you can decide whether or not you desire to participate. You should go ahead: all things considered, as it takes very little time to complete many small internet surveys, you"d not complain about the process if a big prize was won by you in the sketch. <br><br>The chance to make money filling in survey after survey is what most people are seeking, but the industry does not work like that and, so long as you get your fair share of paid studies, you will have the ability to make some good extra money without difficulty. Think about the prize draw surveys as a bonus; if you do not win the prize, you have lost nothing but the few minutes of time it took you to accomplish a online survey and, if you win, you will be getting paid well just for filling out that single short survey. <br><br>If you appear to be getting significantly less than your fair share of invitations to indulge in online surveys for cash, don"t make the mistake of assuming the survey business you have joined only gives reward pull surveys. This telling [http://azdomen.info/blogs/earn-money-o-nline-with-paid-survey/ Earn Money O-nline With Paid Survey | A to Z guide to ab pains] web resource has diverse stirring lessons for when to mull over this hypothesis. It might well be that your demographic report does not match the type of person the study organizations are searching for to accomplish paid studies at that particular time. If you think anything at all, you will seemingly want to check up about [http://journals.fotki.com/coldseeder204/Sideboard-Dinosaur/ this month]. If you are patient, the trend changes will be probably found by you and you begin getting offers to be a part of surveys. You will dsicover that your demographic profile makes you entitled to less reviews. Your eligibility if you are invited to surveys is determined by many factors including gender, age, training, site, occupation, interests, number and age of children and many other activities. <br><br>No body features a report which will fit what"s needed for participation in most settled online survey, and some people are bound to get invited to more survey than others. The way to raise your chances to have paid to take short online surveys for money would be to sign up to as many online review companies as you can find.. This great [http://ritish.info/2014/08/19/are-you-trying-to-find-a-way-to-make-money-hurry-i-will-help-you-2/ visit our site] wiki has a myriad of thought-provoking suggestions for where to allow for this concept.<br><br>Should you liked this informative article and you wish to get details about [http://Dict.leo.org/?search=temporary+health temporary health] insurance; [http://www.getjealous.com/observantaborig60 http://www.getjealous.com], i implore you to go to our web site. |
| In [[mathematics]] and [[computational geometry]], a '''Delaunay triangulation''' for a set '''P''' of points in a plane is a [[triangulation (geometry)|triangulation]] DT('''P''') such that no point in '''P''' is inside the [[Circumcircle#Triangles|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 skinny triangles. The triangulation is named after [[Boris Delaunay]] for his work on this topic from 1934.<ref name="Delaunay1934">B. Delaunay: ''[http://mi.mathnet.ru/eng/izv4937 Sur la sphère vide], Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793–800, 1934''</ref>
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| 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.
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| By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean. However in these cases a Delaunay triangulation is not guaranteed to exist or be unique.
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| ==Relationship with the Voronoi diagram==
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| The Delaunay [[triangulation (geometry)|triangulation]] of a [[discrete space|discrete]] point set '''P''' in [[general position]] corresponds to the [[dual graph]] of the [[Voronoi diagram]] for '''P'''. Special cases include the existence of three points on a line and four points on circle. | |
| <gallery> | |
| File:Delaunay_circumcircles_centers.svg|The Delaunay triangulation with all the circumcircles and their centers (in red).
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| Image:Delaunay_Voronoi.svg|Connecting the centers of the circumcircles produces the [[Voronoi diagram]] (in red).
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| </gallery> | |
| <!-- Is the reader expected to use a microscope to read this article? Can some people see these circles without looking for them? (Rhetorical question. Obviously not.) -->
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| == ''d''-dimensional Delaunay ==
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| For a set '''P''' of points in the (''d''-dimensional) [[Euclidean space]], a '''Delaunay triangulation''' is a [[Triangulation (geometry)|triangulation]] DT('''P''') such that no point in '''P''' is inside the [[circumcircle|circum-hypersphere]] of any [[simplex]] in DT('''P'''). It is known<ref name="deBerg"/> that there exists a unique Delaunay triangulation for '''P''', if '''P''' is a set of points in ''[[general position]]''; that is, there exists no [[flat (geometry)|''k''-flat]] containing ''k'' + 2 points nor a [[n-sphere|''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).
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| 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. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are [[simplex|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.
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| ==Properties==
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| [[File:Example steps in Delauney triangularization.png|thumb|Example steps]] | |
| Let ''n'' be the number of points and ''d'' the number of dimensions.
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| *The union of all simplices in the triangulation is the convex hull of the points.
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| *The Delaunay triangulation contains ''O''(''n''<sup>⌈''d'' / 2⌉</sup>) simplices.<ref>{{cite journal
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| | last = Seidel
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| | first = R.
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| | title = The upper bound theorem for polytopes: an easy proof of its asymptotic version
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| | journal = Computational Geometry
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| | volume = 5
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| | pages = 115–116
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| | year = 1995
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| | url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TYS-3YVD096-C&_user=108429&_coverDate=09%2F30%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000059713&_version=1&_urlVersion=0&_userid=108429&md5=70a4159a39ed8ab2c6709025aa77d5de&searchtype=a
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| | doi = 10.1016/0925-7721(95)00013-Y
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| | issue = 2 }}</ref>
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| *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]]).
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| *In the plane, each vertex has on average six surrounding triangles.
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| *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.<ref>{{citation
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| | last1 = Edelsbrunner | first1 = Herbert | author1-link = Herbert Edelsbrunner
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| | last2 = Tan | first2 = Tiow Seng
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| | last3 = Waupotitsch | first3 = Roman
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| | doi = 10.1137/0913058
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| | issue = 4
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| | journal = SIAM Journal on Scientific and Statistical Computing
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| | mr = 1166172
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| | pages = 994–1008
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| | title = An ''O''(''n''<sup>2</sup> log ''n'') time algorithm for the minmax angle triangulation
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| | volume = 13
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| | year = 1992}}.</ref>
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| *A circle circumscribing any Delaunay triangle does not contain any other input points in its interior.
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| *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.
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| * 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.
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| * The closest neighbor ''b'' to any point ''p'' is on an edge ''bp'' in the Delaunay triangulation since the [[nearest neighbor graph]] is a subgraph of the Delaunay triangulation.
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| * The Delaunay triangulation is a [[geometric spanner]]: the shortest path between two vertices, along Delaunay edges, is known to be no longer than <math>\frac{4\pi}{3\sqrt{3}} \approx 2.418</math> times the Euclidean distance between them.
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| == Visual Delaunay definition: Flipping ==
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| 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.
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| 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:
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| <gallery> | |
| Image:Delaunay_geometry.png|This triangulation does not meet the Delaunay condition (the sum of α and γ is bigger than 180°).
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| Image:Delaunay_before_flip.png|This triangulation does not meet the Delaunay condition (the circumcircles contain more than three points).
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| Image:Delaunay_after_flip.png|''Flipping'' the common edge produces a Delaunay triangulation for the four points.
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| </gallery> | |
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| ==Algorithms==
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| 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]]:<ref>{{cite web| author=Guibas, Lenoidas | coauthors= Stolfi, Jorge |url = http://portal.acm.org/citation.cfm?id=282923 | title = Primitives for the manipulation of general subdivisions and the computation of Voronoi| publisher = ACM | page=107 | date = 1985-04-01| accessdate = 2009-08-01}}</ref>
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| : <math>\begin{vmatrix}
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| A_x & A_y & A_x^2 + A_y^2 & 1\\
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| B_x & B_y & B_x^2 + B_y^2 & 1\\
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| C_x & C_y & C_x^2 + C_y^2 & 1\\
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| D_x & D_y & D_x^2 + D_y^2 & 1
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| \end{vmatrix} = \begin{vmatrix}
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| A_x - D_x & A_y - D_y & (A_x^2 - D_x^2) + (A_y^2 - D_y^2) \\
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| B_x - D_x & B_y - D_y & (B_x^2 - D_x^2) + (B_y^2 - D_y^2) \\
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| C_x - D_x & C_y - D_y & (C_x^2 - D_x^2) + (C_y^2 - D_y^2)
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| \end{vmatrix} > 0
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| </math>
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| When ''A'', ''B'' and ''C'' are sorted in a [[counterclockwise]] order, this determinant is positive if and only if ''D'' lies inside the circumcircle.
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| ===Flip algorithms===
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| 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 O(''n''<sup>2</sup>) edge flips, and does not extend to three dimensions or higher.<ref name="deBerg">{{cite book
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| | last = de Berg
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| | first = Mark
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| | authorlink =
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| | coauthors = Otfried Cheong, Marc van Kreveld, Mark Overmars
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| | title = Computational Geometry: Algorithms and Applications
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| | publisher = Springer-Verlag
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| | year = 2008
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| | id =
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| | url = http://www.cs.uu.nl/geobook/interpolation.pdf
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| | isbn = 978-3-540-77973-5 }}</ref>
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| ===Incremental===
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| 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 naively, 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''<sup>2</sup>).
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| 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.<ref>{{cite journal | |
| | last = Guibas
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| | first = L.
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| | coauthors = D. Knuth; M. Sharir
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| | title = Randomized incremental construction of Delaunay and Voronoi diagrams
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| | journal = Algorithmica
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| | volume = 7
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| | pages = 381–413
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| | year = 1992
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| | url = http://www.springerlink.com/content/p8377h68j82l6860
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| | doi = 10.1007/BF01758770}}</ref>
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| 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.<ref name="deBerg"/> While the technique extends to higher dimension (as proved by Edelsbrunner and Shah<ref>{{cite journal
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| | last = Edelsbrunner
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| | first = Herbert
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| | authorlink = Herbert Edelsbrunner
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| | coauthors = Nimish Shah
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| | title = Incremental Topological Flipping Works for Regular Triangulations
| |
| | journal = Algorithmica
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| | volume = 15
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| | pages = 223–241
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| | year = 1996
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| | url = http://www.springerlink.com/content/4gdja72vx1qmg44x/?p=a74909a339d9498cbff326f08b084b4c&pi=1
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| | doi = 10.1007/BF01975867
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| | issue = 3}}</ref>), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.
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| 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.
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| ===Divide and conquer===
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| A [[divide and conquer algorithm]] for triangulations in two dimensions is due to Lee and Schachter which was improved by [[Leonidas J. Guibas|Guibas]] and [[Jorge Stolfi|Stolfi]]<ref>[http://www.geom.uiuc.edu/~samuelp/del_project.html Computing Constrained Delaunay Triangulations]</ref> 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'').<ref name="Leach1992">{{cite paper | first = G. | last = Leach | title = ''Improving Worst-Case Optimal Delaunay Triangulation Algorithms.'' | id = {{citeseerx|10.1.1.56.2323}} | month = June | year = 1992 }}</ref>
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| 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 worst-case performance.
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| 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>{{cite journal | last = Cignoni | first = P. | coauthors = C. Montani; R. Scopigno | year = 1998 | title = DeWall: A fast divide and conquer Delaunay triangulation algorithm in E<sup>d</sup> | journal = Computer-Aided Design | volume = 30 | issue = 5 | pages = 333–341 | doi = 10.1016/S0010-4485(97)00082-1 }}</ref>
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| Divide and conquer has been shown to be the fastest DT generation technique.<ref>A Comparison of Sequential Delaunay Triangulation Algorithms http://www.cs.berkeley.edu/~jrs/meshpapers/SuDrysdale.pdf</ref><ref>http://www.cs.cmu.edu/~quake/tripaper/triangle2.html</ref>
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| ===Sweepline===
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| [[Fortune's Algorithm]] uses a [[sweepline]] technique to achieve O(''n'' log ''n'') runtime in the planar case.
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| ===Sweephull===
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| Sweephull<ref>[http://www.s-hull.org/paper/s_hull.pdf S-hull]</ref> is a hybrid technique for 2D Delaunay triangulation that uses a radially propagating sweep-hull (sequentially created from
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| the radially sorted set of 2D points, giving a non-overlapping triangulation), paired with a final iterative triangle flipping step.
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| An accurate integer arithmetic variant of the algorithm is also presented.
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| == Applications ==
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| 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.
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| For modelling 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). See [[triangulated irregular network]].
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| Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the [[Delaunay tessellation field estimator|DTFE]].
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| [[Image:Delaunay Triangulation (100 Points).svg|right|thumb|250px|The Delaunay triangulation of a random set of 100 points in a plane.]]
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| 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. 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]].
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| 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 nodal locations so as 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.
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| == See also ==
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| * [[Beta skeleton]]
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| * [[Constrained Delaunay triangulation]]
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| * [[Delaunay tessellation field estimator]]
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| * [[Gabriel graph]]
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| * [[Gradient pattern analysis]]
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| * [[Pitteway triangulation]]
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| * [[Urquhart graph]]
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| * [[Voronoi diagram]]
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| * [[Convex hull algorithms]]
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| * [[Quasitriangulation]]
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| == References ==
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| {{reflist}}
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| == External links ==
| |
| * Delaunay triangulation in [[CGAL]], the Computational Geometry Algorithms Library:
| |
| ** {{cite web
| |
| | last = Yvinec | first = Mariette
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| | title = 2D Triangulation
| |
| | url = http://www.cgal.org/Pkg/Triangulation2
| |
| | accessdate = April 2010
| |
| }}
| |
| ** {{cite web
| |
| | last1 = Pion | first1 = Sylvain
| |
| | last2 = Teillaud | first2 = Monique
| |
| | title = 3D Triangulations
| |
| | url = http://www.cgal.org/Pkg/Triangulation3
| |
| | accessdate = April 2010
| |
| }}
| |
| ** {{cite web
| |
| | last1 = Hert | first = Susan
| |
| | last2 = Seel | first2 = Michael
| |
| | title = dD Convex Hulls and Delaunay Triangulations
| |
| | url = http://www.cgal.org/Pkg/ConvexHullD
| |
| | accessdate = April 2010
| |
| }}
| |
| * {{cite web
| |
| | title = Delaunay triangulation
| |
| | publisher = Wolfram MathWorld
| |
| | url = http://mathworld.wolfram.com/DelaunayTriangulation.html
| |
| | accessdate = April 2010
| |
| }}
| |
| * {{cite web
| |
| | title = Qhull
| |
| | url = http://www.qhull.org
| |
| | accessdate = April 2010
| |
| }} — Code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection
| |
| * {{cite web
| |
| | last = Shewchuk | first = Jonathan Richard
| |
| | title = Triangle
| |
| | url = http://www.cs.cmu.edu/~quake/triangle.html
| |
| | accessdate = April 2010
| |
| }} – A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator
| |
| * {{cite web
| |
| | last1 = Kumar | first1 = Piyush
| |
| | last2 = Mohanty | first2 = Somya
| |
| | title = Triangle++
| |
| | url = http://www.compgeom.com/~piyush/scripts/triangle/
| |
| }} – A C++ wrapper on Triangle
| |
| * {{cite web
| |
| | title = Poly2Tri
| |
| | url = http://code.google.com/p/poly2tri/
| |
| | publisher = Google Code
| |
| | accessdate = April 2010
| |
| }} – A sweepline Constrained Delaunay Triangulation (CDT) library, available in ActionScript 3, C, C++, C#, Go, Haxe, Java, Javascript, Python and Ruby
| |
| | |
| {{DEFAULTSORT:Delaunay Triangulation}}
| |
| [[Category:Triangulation (geometry)]]
| |
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If you appear to be getting significantly less than your fair share of invitations to indulge in online surveys for cash, don"t make the mistake of assuming the survey business you have joined only gives reward pull surveys. This telling Earn Money O-nline With Paid Survey | A to Z guide to ab pains web resource has diverse stirring lessons for when to mull over this hypothesis. It might well be that your demographic report does not match the type of person the study organizations are searching for to accomplish paid studies at that particular time. If you think anything at all, you will seemingly want to check up about this month. If you are patient, the trend changes will be probably found by you and you begin getting offers to be a part of surveys. You will dsicover that your demographic profile makes you entitled to less reviews. Your eligibility if you are invited to surveys is determined by many factors including gender, age, training, site, occupation, interests, number and age of children and many other activities.
No body features a report which will fit what"s needed for participation in most settled online survey, and some people are bound to get invited to more survey than others. The way to raise your chances to have paid to take short online surveys for money would be to sign up to as many online review companies as you can find.. This great visit our site wiki has a myriad of thought-provoking suggestions for where to allow for this concept.
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