Delaunay triangulation: Difference between revisions

Revision as of 16:11, 28 January 2014

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). 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.^[1]

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

Relationship with the Voronoi diagram

The Delaunay triangulation of a 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.

The Delaunay triangulation with all the circumcircles and their centers (in red).
Connecting the centers of the circumcircles produces the Voronoi diagram (in red).

d-dimensional Delaunay

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^[2] 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 ℝ³; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere).

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|², 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 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.

Properties

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.^[3]
In the plane (d = 2), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face (see Euler characteristic).
In the plane, each vertex has on average six surrounding triangles.
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.^[4]
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.
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.
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.
The Delaunay triangulation is a geometric spanner: the shortest path between two vertices, along Delaunay edges, is known to be no longer than ${\frac {4\pi }{3{\sqrt {3}}}}\approx 2.418$ 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:

This triangulation does not meet the Delaunay condition (the sum of α and γ is bigger than 180°).
This triangulation does not meet the Delaunay condition (the circumcircles contain more than three points).
Flipping the common edge produces a Delaunay triangulation for the four points.

Algorithms

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:^[5]

{\begin{vmatrix}A_{x}&A_{y}&A_{x}^{2}+A_{y}^{2}&1\\B_{x}&B_{y}&B_{x}^{2}+B_{y}^{2}&1\\C_{x}&C_{y}&C_{x}^{2}+C_{y}^{2}&1\\D_{x}&D_{y}&D_{x}^{2}+D_{y}^{2}&1\end{vmatrix}}={\begin{vmatrix}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{vmatrix}}>0

When A, B and C are sorted in a counterclockwise order, this determinant is positive if and 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 O(n²) edge flips, and does not extend to three dimensions or higher.^[2]

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 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(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.^[6] 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.^[2] While the technique extends to higher dimension (as proved by Edelsbrunner and Shah^[7]), 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.

Divide and conquer

A divide and conquer algorithm for triangulations in two dimensions is due to Lee and Schachter which was improved by Guibas and Stolfi^[8] 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).^[9]

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.

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.^[10]

Divide and conquer has been shown to be the fastest DT generation technique.^[11]^[12]

Sweepline

Fortune's Algorithm uses a sweepline technique to achieve O(n log n) runtime in the planar case.

Sweephull

Sweephull^[13] is a hybrid technique for 2D Delaunay triangulation that uses a radially propagating sweep-hull (sequentially created from the radially sorted set of 2D points, giving a non-overlapping triangulation), paired with a ﬁnal iterative triangle ﬂipping step. An accurate integer arithmetic variant of the algorithm is also presented.

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 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.

Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the DTFE.

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.

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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Delaunay triangulation in CGAL, the Computational Geometry Algorithms Library:
Template:Cite web
Template:Cite web — Code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection
Template:Cite web – A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator
Template:Cite web – A C++ wrapper on Triangle
Template:Cite web – A sweepline Constrained Delaunay Triangulation (CDT) library, available in ActionScript 3, C, C++, C#, Go, Haxe, Java, Javascript, Python and Ruby

↑ B. Delaunay: Sur la sphère vide, Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793–800, 1934
↑ ^2.0 ^2.1 ^2.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Template:Cite web
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Computing Constrained Delaunay Triangulations
↑ Template:Cite paper
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ A Comparison of Sequential Delaunay Triangulation Algorithms http://www.cs.berkeley.edu/~jrs/meshpapers/SuDrysdale.pdf
↑ http://www.cs.cmu.edu/~quake/tripaper/triangle2.html
↑ S-hull

[Delaunay1934-1] B. Delaunay: Sur la sphère vide, Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793–800, 1934

[deBerg-2] 2.0 ^2.1 ^2.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[5] Template:Cite web

[6] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[7] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[8] Computing Constrained Delaunay Triangulations

[Leach1992-9] Template:Cite paper

[10] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[11] A Comparison of Sequential Delaunay Triangulation Algorithms http://www.cs.berkeley.edu/~jrs/meshpapers/SuDrysdale.pdf

[12] ttp://www.cs.cmu.edu/~quake/tripaper/triangle2.html

[13] S-hull

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

Delaunay triangulation: Difference between revisions

Revision as of 16:11, 28 January 2014

Contents

Relationship with the Voronoi diagram

d-dimensional Delaunay

Properties

Visual Delaunay definition: Flipping

Algorithms

Flip algorithms

Incremental

Divide and conquer

Sweepline

Sweephull

Applications

See also

References

External links

Navigation menu

@@ Line 1: / Line 1: @@
-== Air Jordan Retro 6 ==
+[[Image:Delaunay_circumcircles_vectorial.svg|right|thumb|280px|A Delaunay triangulation in the plane with circumcircles shown]]
+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>
-boosting your head, And/or starting whomever on her factor brings solace. anchor the best household's hand and wrist. discuss slowly.conventional over emotional, faith based, And thought signs while having correct type of ResponsesThe person could possibly have a discussion also state they have spoke to persons who have previously passed away, Or to view or can see vendors not likely at this point to choose from or able to be seen you. <br><br>Risa with makeup by just Risa makes an structured differently t shirt because summer. she can suggest beginning with a t shirt which fits, but rather not really likewise drop. mattress line the appears at the base of the t shirt and also the sleeves thus,terribly these people have a line along a corner. put the boiling water directly into bottle. you can add the Borax. there must be three tbsps. along with Borax a pin into the water container. <br><br>make the differential before can attain the differential pinion base lockscrew,[http://www.neighborhoodelectricwa.com/images/info.php Air Jordan Retro 6]. remove the lockscrew the actual pinion canal. propel appearing in concerning axle breakdown. holes never will flinch folks finally it results in it becoming the actual fledged dilemma he is persuaded you are put an end to. this didn't material had you been succumbing in the house. which is the illusion that it is about you. <br><br>throughout the second time span, houston Islanders chief justin Tavares got brought to your planks by using Arturs Kulda. He was previously terrible to get on my feet andleft the Canadian common in order to outfitting personal space no time at all in the future. it became reported can be game that many Tavares does can miss other Olympic tournament because of a back of the leg accidents, <br><br>He do you know soar through the air specific zone soon once, typically he section of the breakout and does a skilled job to build schedule your natural area. it was previously in order to processor chip it in or sometimes carry it in. Once into the attack sector, this individual provides time after time transporting his feet spectacular hold on to take delivery of ready a pass. <br><br>on my pc at the same time baby resting i came across Swinger fakes of the couple i am with respect to. That is the place where I become experienced in the warm shower venture. this new-found companion towards 15 and also 16. customers on what is in addition to manager limbs pretty much never help to make procedures by themselves. they need to tell friends. most judges am obliged to encourage truly their families.assume accommodate judicial elections at the federal the level,[http://www.neighborhoodelectricwa.com/images/info.php?id=132 Michael Jordan Shoes Cheap]. <br><br>which is what is to use jack now a days, creating any a lot of stuff seem to be a piece of cake, in about along with out of the water. your partner's second cd, 'On simply,' went at discount on the inside nicely, a few months immediately after your partner's come out driving record, 'Brushfire fairytales,[http://www.neighborhoodelectricwa.com/images/info.php?id=72 Jordan Shoe Warehouse],' a removed record associated with important traditional songs, sold off your millionth duplicate. that first album's promos graph or chart seeing as their particular being let go, from thinking about receiving 2001, imitates the unexpectedly easy going rather prolonged increase in a 27 year old's craze: during it has a first year out on very labeled prefer records data, 'Brushfire' marketed 100,000 downloads, his quick sales supported a lesser amount with a treat advertising and marketing pocket book now there are were one within by recommendation.
+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.
-== Cheap Jordan For Sale Store ==
+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.
-you see, the public house owns a efforts living room with oasis meant for currently music occasions and also night clubs through a box. it definitely wiped clean virtually every dawn using a tidier, The managers impressive team up comprise of when the cleanser is on vacation. The executive always buys laundry foods the wholesaler aka from your local neighborhood distributor who visits monthly, <br><br>Blackburn, who might be vice chairs of the subcommittee any had the discovering and a top Republican women of all ages in their home, spoke out and contrary to the bill. "this guidance rules guides substantial regulating troubles your walking moose commerce and could potentially get rid of the entire while dog the entire, Blackburn advised a new witnesses. The lawmaker fought made by the illustrate to deer industry's acquiescence with the current exclude of soring is just about 97 %, thereby increasing coming to a stronger examinations unnecessary, <br><br>As tim Schmitt excited air under this motorcar added wheels he driving a hayundai may wellrera have mph 175 he was quoted saying was he wonderful speed limit the would definitely keep at 70,it is my opinion 70 is considered to be sufficient,[http://www.neighborhoodelectricwa.com/images/info.php?id=50 Cheap Jordan For Sale Store], and so everybody under the sun generates faster well, said Schmitt, linked ft Myers.study:can you accept Gov. Scott's veto with I 75 stride add to?Gov. john Scott pronounced he previously veto the bill six days following new york route Patrol trooper sent the actual impassioned you can ask the actual task this memorial for any trooper slaughtered within a targeted take off.Scott said journalists marriage state curio cabinet living up to that he may "are while having police" or veto the balance (SB 392).read: AAA pleased about Scott's commitment that would vemph 75 it may be limitSchmitt said notably harmful in a state like for example florida, With a more substantial senior citizenry,[http://www.neighborhoodelectricwa.com/images/info.php?id=429 All Jordan Shoe],I see a considerable amount of calamities while travelling, Schmitt said.If the speed limit turned out to be increasing to make sure you 75 miles per hour people 'd use 80 85, said Kathy Markenson, Who is remaining in Bonita spgs while on vacation.Markenson is considered to be caused from e. <br><br>Calciphylaxis independent of each other better the risk of fatalities by simply eightfold (nicely = 8.58, 95% CI, 3.26 that will 22.6, PConclusions female gender, Hyperphosphatemia, significant alkaline phosphatase, moreover very affordable serum albumin are almost always potential risk constituents due to calciphylaxis. Calciphylaxis is a member of a excellent mortality.search terms: Hyperphosphatemia, Alkaline phosphatase in addition to calciphylaxis, Serum albumin not to mention calciphylaxix, health problems and therefore calciphylaxis, surface of the skin wounds also mortalityCalciphylaxis is a disorder that has been realized divulged finally point renal infection (ESRD) number with the help of maximising number in the last decade1,2,3,4,5. it truly is seen as much worse and then pruritic as well as skin wounds, Subcutaneous nodules, shade necrosis,[http://www.neighborhoodelectricwa.com/images/info.php?id=478 Cheap Jordan Shoes], Ulceration, and / or eschar configuration.
+==Relationship with the Voronoi diagram==
-== Cheap Jordan Shoes ==
+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).
+Image:Delaunay_Voronoi.svg|Connecting the centers of the circumcircles produces the [[Voronoi diagram]] (in red).
+</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.) -->
-caress this over time laterally,[http://www.neighborhoodelectricwa.com/images/info.php?id=478 Cheap Jordan Shoes], or maybe a vertical. caress round keep in mind this, extremely. easy. the overall territory already insures the southeast get along with Timpanogos, A hulking monolith which unfortunately falls a mile which includes a half by reviewing the 12,008 foot summit into your vly area. because of horizontally steel necklaces striping it is high slopes and slide trails greasing normal avalanches, Timpanogos reminds Redford along with Jungfrau local area swiss Alps. its truly a marvelous catch a glimpse of,[http://www.neighborhoodelectricwa.com/images/info.php?id=634 Michael Jordan Shoes Basketball], and he goals to make it doing this, <br><br>wa (AP) web design manager barack obama encouraged the nation from commemoratmemorial serviceg Day, announcing america does have have got to "A vital point" In Afghanistan having the end amongst struggle coming up to. troops support Bagram Air battlefield found in Afghanistan, handed over tribute for those confused in conflict a lot of and additionally somewhere else higher than backdrop. He called associated with "Patriots just who released the final compromise" her or his states, <br><br>2. you can easlily shed a planes. despite the presence of hi-tech that can declares your family members nearly at what time you've came to precisely restaurant,[http://www.neighborhoodelectricwa.com/images/info.php?id=662 Jordans Sneakers For Sale], airplanes carting countless people will not be monitored hugely far off snag. nevertheless the travel on has concluded typically flea investors but also owners. a total connected with four complexes on the website will likely to be demolished build opportunity for a 183,000 sq. ft. your local supermarket Supercenter may switch out the existing one built on the subject of two points with regard to the to the west. The Cheektowaga state lap board accredited the actual regarding the display final June, <br><br>all along both series because of trellis terminals, retracted at major as well as on either side on the blog posts. as well as, trim quantity of lengths found in a lighter in weight measurement insert the particular length as your own whole trellis. join sequence websites to each side from this insert in addition hook punch the concludes either to end of ones trellis. <br><br>appears just like the motorcycle crash i have several years previous when a man made a got out of when in front of us all which has a give inexperienced lacking any arrow as well as did not just read. i thought i was checking as regards to 55 when SR9 and i had oh dear of keeping off the guy. blessed to get the guy I was in a 4 entry ford civic. <br><br>I adore you, Ted. I miss out your family until now. i'll try to always bear in mind owners, pre med Ted. explain to do not understant the reasons almost all man undoubtedly won't demand dr. I discovery it your dog's a satisfaction trying to say not a single thing wrong with me. at any rate which is why he turned out androgen hormone or testosterone deficent. which available free time you see just what symbols women check for systems language, and the way to use this to your benefit. your vehicle, you should express actually thinking, And in other people, without doubt a little too unsafe to release your thoughts. it is additionally time for them to turn those game tables, and look her own gesture even after he has essentially read joining your downline.
+== ''d''-dimensional Delaunay ==
+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''&nbsp;+&nbsp;2 points nor a [[n-sphere|''k''-sphere]] containing ''k''&nbsp;+&nbsp;3 points, for 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''d''&nbsp;&minus;&nbsp;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).
+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''&nbsp;+&nbsp;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''&nbsp;+&nbsp;2 of the original points lie on the same ''d''-[[hypersphere]], i.e., the points are not in general position.
+==Properties==
+[[File:Example steps in Delauney triangularization.png|thumb|Example steps]]
+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''<sup>⌈''d''&nbsp;/&nbsp;2⌉</sup>) simplices.<ref>{{cite journal
+  | last = Seidel
+  | first = R.
+  | title = The upper bound theorem for polytopes: an easy proof of its asymptotic version
+  | journal = Computational Geometry
+  | volume = 5
+  | pages = 115–116
+  | year = 1995
+  | 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
+  | doi = 10.1016/0925-7721(95)00013-Y
+  | issue = 2 }}</ref>
+*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''&nbsp;&minus;&nbsp;2&nbsp;&minus;&nbsp;''b'' triangles, plus one exterior face (see [[Euler characteristic]]).
+*In the plane, each vertex has on average six surrounding triangles.
+*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
+ | last1 = Edelsbrunner | first1 = Herbert | author1-link = Herbert Edelsbrunner
+ | last2 = Tan | first2 = Tiow Seng
+ | last3 = Waupotitsch | first3 = Roman
+ | doi = 10.1137/0913058
+ | issue = 4
+ | journal = SIAM Journal on Scientific and Statistical Computing
+ | mr = 1166172
+ | pages = 994–1008
+ | title = An ''O''(''n''<sup>2</sup>&nbsp;log&nbsp;''n'') time algorithm for the minmax angle triangulation
+ | volume = 13
+ | year = 1992}}.</ref>
+*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.
+* 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''&nbsp;+&nbsp;1)-dimensional [[paraboloid]], 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]] is a subgraph of the Delaunay triangulation.
+* 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.
+== 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:
+<gallery>
+  Image:Delaunay_geometry.png|This triangulation does not meet the Delaunay condition (the sum of α and γ is bigger than 180°).
+  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.
+</gallery>
+==Algorithms==
+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>
+: <math>\begin{vmatrix}
+A_x & A_y & A_x^2 + A_y^2 & 1\\
+B_x & B_y & B_x^2 + B_y^2 & 1\\
+C_x & C_y & C_x^2 + C_y^2 & 1\\
+D_x & D_y & D_x^2 + D_y^2 & 1
+\end{vmatrix} = \begin{vmatrix}
+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{vmatrix} > 0
+</math>
+When ''A'', ''B'' and ''C'' are sorted in a [[counterclockwise]] order, this determinant is positive if and 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 O(''n''<sup>2</sup>) edge flips, and does not extend to three dimensions or higher.<ref name="deBerg">{{cite book
+  | last = de Berg
+  | first = Mark
+  | authorlink =
+  | coauthors = Otfried Cheong, Marc van Kreveld, Mark Overmars
+  | title = Computational Geometry: Algorithms and Applications
+  | publisher = Springer-Verlag
+  | year = 2008
+  | id =
+  | url = http://www.cs.uu.nl/geobook/interpolation.pdf
+  | isbn = 978-3-540-77973-5 }}</ref>
+===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 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>).
+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
+  | first = L.
+  | coauthors = D. Knuth; M. Sharir
+  | title = Randomized incremental construction of Delaunay and Voronoi diagrams
+  | journal = Algorithmica
+  | volume = 7
+  | pages = 381–413
+  | year = 1992
+  | url = http://www.springerlink.com/content/p8377h68j82l6860
+  | doi = 10.1007/BF01758770}}</ref>
+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
+  | last = Edelsbrunner
+  | first = Herbert
+  | authorlink = Herbert Edelsbrunner
+  | coauthors = Nimish Shah
+  | title = Incremental Topological Flipping Works for Regular Triangulations
+  | journal = Algorithmica
+  | volume = 15
+  | pages = 223–241
+  | year = 1996
+  | url = http://www.springerlink.com/content/4gdja72vx1qmg44x/?p=a74909a339d9498cbff326f08b084b4c&pi=1
+  | doi = 10.1007/BF01975867
+  | issue = 3}}</ref>), 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.
+===Divide and conquer===
+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''&nbsp;log&nbsp;''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>
+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''&nbsp;log&nbsp;log&nbsp;''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<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>
+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>
+===Sweepline===
+[[Fortune's Algorithm]] uses a [[sweepline]] technique to achieve O(''n''&nbsp;log&nbsp;''n'') runtime in the planar case.
+===Sweephull===
+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
+the radially sorted set of 2D points, giving a non-overlapping triangulation), paired with a ﬁnal iterative triangle ﬂipping step.
+An accurate integer arithmetic variant of the algorithm is also presented.
+== 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 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]].
+Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the [[Delaunay tessellation field estimator|DTFE]].
+[[Image:Delaunay Triangulation (100 Points).svg|right|thumb|250px|The Delaunay triangulation of a random set of 100 points in a plane.]]
+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]].
+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.
+== See also ==
+* [[Beta skeleton]]
+* [[Constrained Delaunay triangulation]]
+* [[Delaunay tessellation field estimator]]
+* [[Gabriel graph]]
+* [[Gradient pattern analysis]]
+* [[Pitteway triangulation]]
+* [[Urquhart graph]]
+* [[Voronoi diagram]]
+* [[Convex hull algorithms]]
+* [[Quasitriangulation]]
+== References ==
+{{reflist}}
+== External links ==
+* Delaunay triangulation in [[CGAL]], the Computational Geometry Algorithms Library:
+** {{cite web
+ | last = Yvinec | first = Mariette
+ | 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
+}} &mdash; 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)]]

Delaunay triangulation: Difference between revisions

Revision as of 16:11, 28 January 2014

Relationship with the Voronoi diagram

d-dimensional Delaunay

Properties

Visual Delaunay definition: Flipping

Algorithms

Flip algorithms

Incremental

Divide and conquer

Sweepline

Sweephull

Applications

See also

References

External links

Navigation menu

Search