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Topic: Delaunay triangulation

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	Boris Delaunay - Wikipedia, the free encyclopedia
		Boris Nikolaevich Delaunay (March 15, 1890 – July 17, 1980) (or Delone; Russian language: Борис Николаевич Делоне), was a Soviet/Russian mathematician.
		The Delaunay triangulation is named in his honor.
		Delaunay is a French-style transcription of his name that he himself preferred because his French ancestors spelled their name that way.)
	en.wikipedia.org /wiki/Boris_Nikolaevich_Delaunay (148 words)

	Delaunay triangulation (Site not responding. Last check: 2007-10-06)
		In mathematics and computational geometry the Delaunay triangulation for a set P of points the plane is the triangulation DT(P) of P such that no in P is inside the circumcircle of any triangle in DT(P).
		Equivalently the Delaunay triangulation of a discrete point set P is the dual of the Voronoi tessellation for P.
		It is known that the Delaunay triangulation and is unique for P if P a set of points in general position i.e.
	www.freeglossary.com /Delaunay_triangulation (594 words)

	3.2.4 -Delaunay triangulation (Site not responding. Last check: 2007-10-06)
		In a constrained Delaunay triangulation the pre-defined edges are in the triangulation and the empty circumcircle property is modified to apply only to points that can be `seen' from at least one node of the triangle, where the pre-defined edges are treated as opaque.
		The alternative to constrained triangulation is to make the Delaunay triangulation conform to the boundary curve/surface (rather than the boundary discretisation) by adding enough points on the boundary to ensure that the Delaunay triangulation will include the (small) edges on the surface.
		Delaunay triangulation is the fastest method for unstructured mesh generation but boundary conformance needs to be checked/maintained.
	www.epcc.ed.ac.uk /overview/publications/training_material/tech_watch/96_tw/tw-meshgen/MeshGeneration.book_29.html (1259 words)

	SYCODE - Delaunay Triangulation
		A triangulation is the division of a set of points into a set of triangles, usually with the restriction that each triangle side is entirely shared by two adjacent triangles.
		A Delaunay Triangulation of a set of points is a triangulation with the property that every edge is contained in a circle that contains no other points of the set.
		One would hope that the Delaunay Triangulation would be the result of repeatedly joining the two closest points unless doing so would cross an edge you already have, but sadly that turns out not to be the case.
	www.sycode.com /others/delaunay_triangulation.htm (177 words)

	Delaunay Tessellation
		Delaunay tessellation (the Delaunay triangulation in the plane) is another fundamental computational geometry structure.
		The Delaunay tessellation is a “dual tessellation” of the Voronoi diagram.
		The Delaunay triangulation has been extensively studied in the literature and following is a list of important properties of the DT obtained under Assumption 1 (above).
	www.personal.kent.edu /~rmuhamma/Compgeometry/MyCG/CG-Applets/DelaTessel/delacli.htm (452 words)

	Delaunay triangulation (Site not responding. Last check: 2007-10-06)
		In mathematics, and computational geometry, the Delaunay triangulation, for a set P of points in theplane, is the triangulation DT(P) of P such that no point in P is insidethe circumcircle of any triangle in DT(P).
		For a set P of points in the (n-dimensional) Euclidean space, theDelaunay triangulation is the triangulation DT(P) of Psuch that no point in P is inside the circum-hypersphere of any simplex in DT(P).
		The Delaunay triangulation of a discrete point set P is the dual of the Voronoitessellation for P.
	www.therfcc.org /delaunay-triangulation-186272.html (284 words)

	Delaunay Triangulation Method (Site not responding. Last check: 2007-10-06)
		Delaunay triangulation is one of the the most popular methods for generation of unstructured meshes.
		The Delaunay Triangulation of a point set is defined by the empty circle condition, i.e, the triangle is a valid triangle if and only if its circumcircle encloses no other points of the point set.
		In 2D only, of all triangulations, the Delaunay triangulation maximizes the minimum angle for all triangular elements which is the requirement for good quality finite elements.
	www.inf.ethz.ch /personal/cetin/thesis/thesis/node25.html (347 words)

	qdelaunay -- Delaunay triangulation
		The Delaunay triangulation is the triangulation with empty circumspheres.
		Each region of the Delaunay triangulation corresponds to a facet of the lower half of the convex hull.
		Delaunay triangulations do not include facets that are coplanar with the convex hull of the input sites.
	www.qhull.org /html/qdelaun.htm (1835 words)

	Voronoi/Delaunay Applet (Site not responding. Last check: 2007-10-06)
		The Delaunay Triangulation is the geometric dual of the Voronoi Diagram.
		The Voronoi Diagram is built on-the-fly from the Delaunay Triangulation.
		The Delaunay Triangulation is built within a large triangle whose vertices are well off-screen.
	www.cs.cornell.edu /Info/People/chew/Delaunay.html (467 words)

	Delaunay triangulation - The Jiggies Reference Guide (Site not responding. Last check: 2007-10-06)
		In mathematics, and computational geometry, the Delaunay triangulation, for a set P of points in the plane, is the triangulation DT(P) of P such that no point in P is inside the circumcircle of any triangle in DT(P).
		The Delaunay triangulation of a discrete point set P is the dual of the Voronoi tessellation for P.
		A facet not being a simplex implies that n+2 of the original points lay on the same d-hypersphere, and the points were not in general position.
	www.jiggies.com /reference/Delaunay_triangulation (300 words)

	Parallel Delaunay Triangulation (Site not responding. Last check: 2007-10-06)
		The popularity of Delaunay triangulation is that the algorithm produces the most equiangular triangles of all possible methods and it can be computed in O(n·log(n)) in the worst case (where n is the number of points to triangulate).
		Due to this popularity, no wonder that construction of the Delaunay triangulation is one of the problems that are tried to be solved in parallel or distributed environment.
		Triangulation DT(P) of a set of points P in the plane is a Delaunay triangulation of P if and only if the circumcircle of any triangle of DT(P) does not contain any other point of P in its interior.
	www.cg.tuwien.ac.at /studentwork/CESCG/CESCG-2001/JKohout/index.html (3292 words)

	Triangle: Definitions (Site not responding. Last check: 2007-10-06)
		A Delaunay triangulation of a vertex set is a triangulation of the vertex set with the property that no vertex in the vertex set falls in the interior of the circumcircle (circle that passes through all three vertices) of any triangle in the triangulation.
		A constrained Delaunay triangulation of a PSLG is similar to a Delaunay triangulation, but each PSLG segment is present as a single edge in the triangulation.
		A conforming Delaunay triangulation (CDT) of a PSLG is a true Delaunay triangulation in which each PSLG segment may have been subdivided into several edges by the insertion of additional vertices, called Steiner points.
	www.cs.cmu.edu /~quake/triangle.defs.html (302 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		A constrained Delaunay triangulation is a Delaunay triangulation of a set of points and straight-line segments.
		In the paper the definition of constrained Delaunay triangulation is introduced and its basic properties are discussed.
		A new on-line algorithm for constrained Delaunay triangulation is proposed, which is based on the stepwise refinement of an existing triangulation by the incremental insertion of points and constraint segments.
	www.disi.unige.it /person/PuppoE/abstracts/gmip92.html (114 words)

	Applications of 3D Delaunay triangulation algorithms in geoscientific modelling.
		In this research, Delaunay triangulation procedures have been used in the reconstruction of 3D geometric figures where the complexity of the problem is much greater than the 2D case.
		The Delaunay triangulation for a set of points is defined only if the points are distinct, so the uniqueness of the points in the given set is treated as a prerequisite for the applicability of this algorithm.
		For a given set of points in two dimensions, the Delaunay triangulation is univocaly determined and therefore unique, but there are some cases when the triangulation is not unique as there exist different ways of connecting points and all lead to a valid triangulation.
	www.ncgia.ucsb.edu /conf/SANTA_FE_CD-ROM/sf_papers/lattuada_roberto/paper.html (3508 words)

	SDI : Fast & Fully Dynamic Constrained Delaunay Triangulation Library by David Kornmann (Site not responding. Last check: 2007-10-06)
		Usually, other triangulation libraries that you will find will require their own data formats, forcing you to have to deal with multiple databases basically representing the same thing.
		With SDI, it is possible to share the same triangulation database between SDI for the geometry update, the rendering part of your application, and other specific computations, saving them a lot of memory.
		The memory management is static, which means that all the memory needed for the triangulation sessions is allocated at once when the user defined data is connected to the triangulator.
	www.dlc.fi /~dkpa (1359 words)

	Centre of Computer Graphics and Data Visualisation
		Delaunay triangulation is one of the fundamental topics in the computational geometry and it is used in many areas, such as terrain modeling (GIS), scientific data visualization and interpolation, robotics, pattern recognition, meshing for finite element methods (FEM), natural sciences, computer graphics and multimedia, etc.
		We have developped several parallel methods for the construction of the Delaunay triangulation based on randomized incremental insertion with local improvements.
		The first one was to speed-up the computation of the Delaunay triangulation and, the second one, to process large data sets.
	herakles.zcu.cz /research/pdt/index.php (929 words)

	Citations: Voronoi Diagrams from Convex Hulls - Brown (ResearchIndex) (Site not responding. Last check: 2007-10-06)
		....Delaunay triangulation in R d can be computed from a convex hull in R d 1.
		To determine the Delaunay triangulation of a set of points: lift the points to a paraboloid and compute their convex hull.
		By the correspondence between Delaunay triangulation and convex hull, each triangle is a facet of the convex hull and the in sphere test determines the visible facets for the lifted point
	citeseer.ist.psu.edu /context/25875/0 (1848 words)

	Investigating Delaunay Triangulation as a Basis for a Traveling Salesman Problem Algorithm
		Delaunay Triangulation (DT) is a method that has not been fully investigated, though it is known that the DT does not always contain the solution.
		A Delaunay Triangulation of a point set is a triangulation of the point set with the property that no point falls in the interior of the circumcircle (a circle that passes through all three vertices) of any other triangle.
		In the case of the triangulation, the objects are the centers of each triangle.
	oas.ucok.edu /OJAS/00/Papers/phan.htm (2054 words)

	Constrained Delaunay Triangulation using Plane Subdivision (Site not responding. Last check: 2007-10-06)
		Triangulation of set of points on the plane presents a partition of an area, surrounded by a convex hull.
		A succesive refinement of the Delaunay triangulation was described by Ruppert [6], where additional Steiner points are used.
		Triangulation of pseudo-polygons (see pseudocode in Figure 10) is based on strategy "divide and conquer" (Figure 8).
	www.cg.tuwien.ac.at /studentwork/CESCG/CESCG/web/Domiter-Vid (2354 words)

	Triangulate (Site not responding. Last check: 2007-10-06)
		Triangulation involves creating from the sample points a set of non-overlapping triangularly bounded facets, the vertices of the triangles are the input sample points.
		There are a number of triangulation algorithms that may be advocated, the more popular algorithms are the radial sweep method and the Watson algorithm which implement Delaunay triangulation.
		Triangulation based methods honor this situation by giving a large number of triangles and hence more detail to the highly sampled regions and large triangles, less detail, to the regions with a few samples.
	astronomy.swin.edu.au /~pbourke/terrain/triangulate (1866 words)

	Appendix 6 (Site not responding. Last check: 2007-10-06)
		Delaunay triangulation is a method or partitioning known information at irregularly distributed data points to allow smooth, local interpolation between the points.
		It uses the method of natural neighbor interpolation, meaning that the value of an arbitrary point is determined by the three closest points and the relative distance from each.
		A Delaunay triangulation can be found using the empty circle method, which says that the circle passing through three natural neighbor points will contain no other data points.
	www.cs.earlham.edu /~jimg/mcm/appendix6.html (209 words)

	Triangulation
		Implementations: Triangle, by Jonathan Shewchuk of Carnegie-Mellon University, is a C language code that generates Delaunay triangulations, constrained Delaunay triangulations (forced to have certain edges), and quality-conforming Delaunay triangulations (which avoid small angles by inserting extra points).
		Higher-dimensional Delaunay triangulations are a special case of higher-dimensional convex hulls, and Qhull [BDH97] appears to be the convex hull code of choice for general dimensions (i.e.
		Linear-time algorithms for triangulating monotone polygons have been long known [GJPT78] and are the basis of algorithms for triangulating simple polygons.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE186.HTM (1143 words)

	Isosurfaces and Delaunay triangulation (Site not responding. Last check: 2007-10-06)
		The Delaunay triangulation of a set of points V in the Euclidean space is the set of triangles in 2D (or tetrahedra in 3D) whose circumscribed balls contain no points of V in their interior.
		We say that a triangulated surface satisfies the Delaunay constraint if this surface is a subset of the Delaunay triangulation of its vertices.
		Unlike existing methods, our technique builds iso-surfaces that are included in the Delaunay triangulation of their vertices.
	www.labri.fr /Perso/~lachaud/Research/isosurface-Delaunay-en.html (147 words)

	Citations: Delaunay triangulation and the convex hull of n points in expected linear time - Maus (ResearchIndex) (Site not responding. Last check: 2007-10-06)
		Delaunay triangulation and the convex hull of n points in expected linear time.
		In this section, we describe the Delaunay triangulation and a fast method for its construction, following [44] 10 4.2 De nitions and data structures The Delaunay triangulation can be (and historically has been) de ned in many ways.
		In this section, we describe the Delaunay triangulation and a fast method for its construction, following [44] 4.2 Definitions and data structures The Delaunay triangulation can be (and historically has been) defined in many ways.
	citeseer.ist.psu.edu /context/438924/0 (1701 words)

	Reference Manual: CGAL_Delaunay_triangulation_2
		A Delaunay triangulation of a set of points is a triangulation of the sets of point that fulfills the following empty circle property (also called Delaunay property): the circumscribing circle of each face does not contain any vertex.
		On the contrary, because the concept of Delaunay triangulation relies on the notions of empty circles and of distance, the geometric traits is required to provide some additional predicates namely the famous in_circle test and also a Distance class.
		However, the user of an exotic metric must be carefull that the constructed triangulation has to be a triangulation of the convex hull which means that convex hull edges have to be Delaunay edges.
	graphics.stanford.edu /courses/cs368/CGAL/ref-manual2/Triangulation/CGAL_Delaunay_triangulation_2.html (861 words)

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