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Topic: Polygon triangulation

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	Linkages (Part II) - Old Wine in New Bottles
		It is not surprising that triangulations of the plane would be of considerable interest because among all graphs that can be drawn in the plane with a fixed number of vertices, these graphs have the maximal number of edges.
		Triangulations are also known as maximal planar graphs because of the property that if one adds even a single edge between two vertices of a triangulation that are not already joined, the result is a graph which can not be redrawn in the plane without crossings at points that are not vertices.
		triangulations, near triangulations, triangulated polygons) so that they can see what properties all the structures have and what properties fail to hold due to the small changes in the definitions of the structures.
	www.ams.org /featurecolumn/archive/links4.html (1294 words)

	[No title]
		We begin by introducing a triangulation of P. triangulation of a simple polygon is a planar subdivision of (the interior of) P whose vertices are the vertices of P and whose faces are all triangles.
		An important concept in polygon triangulation is the notion of a diagonal, that is, a line segment between two vertices of P that are visible to one another.
		Monotone Polygons: A polygonal chain C is said to be strictly monotone with respect to a given line L, if any line that is orthogonal to L intersects C in at most one point.
	www.cs.wustl.edu /~pless/546/lectures/l6.html (2841 words)

	Polygon Triangulation
		In computer graphics, polygon triangulation algorithms are widely used for tessellating curved geometries, as are described by splines [Kumar and Manocha 1994].
		A monotone polygon is a polygon whose boundary consists of two y-monotone chains.
		A monotone polygon can be triangulated in linear time by using a simple greedy algorithm which repeatedly cuts off the convex corners of the polygon [Fournier and Montuno 1984].
	www.cs.unc.edu /~dm/CODE/GEM/chapter.html (837 words)

	CG & CG Lab (Director: Martin Held) - FIST: Fast Industrial-Strength Triangulation
		Triangulating a polygon also is a fundamental operation in computational geometry, and it has received wide-spread interest over the last two decades.
		Virtually all published triangulation algorithms assume that the polygon is simple, i.e., that the vertices of the polygon are the only points of the plane that belong to two edges, and that no point of the plane belongs to more than two edges.
		Insert every point of intersection among the contours as a new Steiner vertex, compute a conforming triangulation of the convex hull of all vertices, and discard those triangles that lie in the "exterior" of the area specified by the polygons, where an appropriate notion of "interior/exterior" has to be defined.
	www.cosy.sbg.ac.at /~held/projects/triang/triang.html (1907 words)

	Citations: Polynomial-size nonobtuse triangulation of polygons - Bern, Eppstein (ResearchIndex)
		....triangulations of planar graphs is due to Chew [6] which starts with the DT of the input and adds a steiner point at the center of chosen triangles.
		Another problem is that the Delaunay triangulation of n points in E 3 may have size O(n 2) It is natural to ask whether we can add points to the set to reduce the size of its Delaunay triangulation.
		Polynomial-Size Nonobtuse Triangulation Of Polygons - Bern, Eppstein (1992)
	citeseer.ist.psu.edu /context/24544/317304 (2940 words)

	Citations: Minimal triangulations of Polygonal Domains - Klincsek (ResearchIndex)
		If the polygon is convex, a triangulation of weight O(log n) times the polygon s perimeter can be found by the ring heuristic of repeatedly connecting all pairs of adjacent even numbered vertices [21] and as mentioned above a constant factor approximation to the MWT can be computed in linear....
		To obtain the optimal triangulation for P, we record the k that is chosen to optimize each L[i; i j] and use it later to trace the triangulation.
		A simplified variant of the minimum weight triangulation problem is to find a minimum weight triangulation of a polygon P, MWT(P) That is, find a set of diagonals and edges of P such that the sum of the weights of the diagonals and edges is minimum.
	citeseer.ist.psu.edu /context/142375/0 (2323 words)

	ICS 260: Lecture notes on optimal triangulation
		The goal would then be to find a triangulation in which every triangle has quality greater than some threshhold Q, and in which Q is as large as possible; that is, we are trying to maximize the minimum quality of any triangle.
		However, for simple polygons (polygons without holes, in which the triangulation vertices are only allowed to lie on polygon vertices) both of these optimal triangulation problems can be solved by dynamic programming.
		For the case of the max-min-quality triangulation, we can measure the quality of a regular polygon triangle by examining the quality of the corresponding triangle in the input, however if a triangle would be disallowed because it crosses the input polygon's boundary, we give the corresponding regular polygon triangle quality zero.
	www.ics.uci.edu /~eppstein/260/011023 (1756 words)

	Polygon Triangulation in C# - The Code Project - C# Programming
		Polygon is one of the most important objects we are dealing with when we rendering 2D/3D graphics or processing computational geometry.
		As a polygon could be very complicated, some restrictions may be applied on implementation.
		A simple polygon is a polygon with no two non-consecutive edges intersecting.
	www.codeproject.com /csharp/cspolygontriangulation.asp (688 words)

	Triangulation -- from Wolfram MathWorld
		Triangulation is the division of a surface or plane polygon into a set of triangles, usually with the restriction that each
		It was proved in 1925 that every surface has a triangulation, but it might require an infinite number of triangles and the proof is difficult (Francis and Weeks 1999).
		A surface with a finite number of triangles in its triangulation is called compact.
	mathworld.wolfram.com /Triangulation.html (344 words)

	Polygon triangulation . Bernard Chazelle . 1991 (Site not responding. Last check: 2007-10-29)
		Polygon triangulation is a topic in computational geometry.
		A triangulation of a polygon P is its partition into non-overlapping triangles whose union is P.
		A convex polygon is trivial to triangulate in linear time, by adding edges from one vertex to all other vertices.
	www.uk.fraquisanto.net /Polygon_triangulation (402 words)

	Simple polygon . Polygon triangulation . Point in polygon . Bernard Chazelle (Site not responding. Last check: 2007-10-29)
		These are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it.
		A polygon that is not simple is a complex polygon, and does not always have a well-defined inside and outside.
		A triangulation of a polygon P is its partition into...
	www.uk.fraquisanto.net /Simple_polygon (523 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		A triangulation of a simple polygon consists of n-3 non-intersecting diagonals or n-2 triangles where n is the number of vertices in the simple polygon.
		The problem of triangulating a polygon can be state as: given a sequence of N points (specified in either clockwise or counterclockwise order), find N-3 diagonals that partition this polygon into N-2 triangles (usually not unique).
		The basic ideas of this algorithm to triangulate the polygon are: Firstly, trying to find an ear of a polygon, then cut if off from the polygon; that is, a diagonal is added to thr triangulation and a vertex is deleted from the polygon.
	www.mema.ucl.ac.be /~wu/FSA2716-2002/project.html (1423 words)

	Triangulation - Definition, explanation
		In trigonometry and elementary geometry, triangulation is the process of finding a distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other reference points.
		Triangulation is used for many purposes, including surveying, navigation, astrometry, binocular vision and gun direction of weapons.
		Complex triangulation problems involving real-world observations with errors require the solution of large systems of simultaneous equations to generate solutions.
	www.calsky.com /lexikon/en/txt/t/tr/triangulation.php (517 words)

	Computational Geometry Project: The Hertel-Mehlhorn Algorithm
		Polygon partitioning is an important preprocessing step for many geometric algorithms, because most geometric problems are simpler and faster on convex objects than on non-convex ones.
		The "mother" of all polygon partitioning problems is triangulation, where the interior of the polygon is completely partitioned into triangles.
		Two types of partition of a polygon P may be distinguished: a partition by diagonals or a partition by segments.
	www.bringyou.to /compgeom (1575 words)

	Polygon Partitioning
		The process of triangulation of a polygon P consists of finding diagonals within polygon P. Note that a diagonal is a segment between two vertices of polygon P whose interior does not intersect the boundary of P. That is to say, both vertices are visible from each other.
		The triangulation is not deterministic, but it is certainly possible to show that every triangulation of a polygon P of n vertices has n-3 diagonals and results in n-2 triangles.
		In this next section we will show that monotone polygons can be linearly triangulated so partitioning a simple polygon into monotone polygon is the key to efficient triangulation algorithms.
	www.personal.kent.edu /~rmuhamma/Compgeometry/MyCG/PolyPart/polyPartition.htm (1241 words)

	Polygon Triangulation
		Note that three consecutive points of a polygon do not necessarily form an ear - if the line segment between the first and third point intersect with the border of the polygon, or if this line segment lies outside the polygon completely, the three consecutive points do not form an ear.
		The simple, brute-force method of finding an ear of a polygon is to first test the first, second and third points of the polygon to see if they form an ear; if they don't, then test the second, third and fourth points, etc. until an ear is found.
		A faster (and more complicated) method involves dividing a polygon into progressively smaller pieces, until an ear is found or until the polygon is reduced to a single triangle, whichever comes first.
	www.sfu.ca /~cjs/triangulation (285 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-10-29)
		One way to triangulate a simple polygon is by using the assertion that any simple polygon without holes has at least two so called 'ears'.
		A monotone polygon is one with a boundary that consists of two parts, each of which consists of points that have incrementing coordinates in one dimension.
		Such a polygon can easily be triangulated in linear time as described by A.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Polygon_triangulation (447 words)

	Triangulations to Quadrangulations
		Since t=n-2 for the dual tree T of the triangulated polygon T, the most number of unmatched nodes in the dual tree of an n-gon is at most floor(n/3).
		A decomposition of a polygon, P, such that the finite elements are quadrilaterals (as opposed to triangles).
		Given a triangulated domain, T, it can be three-colored if the vertices of the domain can be colored with THREE colors such that no triangle in the domain is assigned the same color to two of its vertices.
	www.cs.mcgill.ca /~cs507/projects/1998/rachelp (3190 words)

	Fast Polygon Triangulation based on Seidel's Algorithm
		These polygons are computed from the trapezoidal decomposition by checking whether the two vertices of the original polygon lie on the same side.
		The triangulation code is invoked through the main interface routine, int triangulate_polygon(n, vertices, triangles); with an n-sided polygon given for input (the vertices are specified in canonical anticlockwise order with no duplicate points).
		Once the triangulated, point-location queries can be invoked as int is_point_inside_polygon(vertex); additional details appear in the C source code which accompanies this Gem.
	www.whisqu.se /per/docs/math13.htm (946 words)

	[No title]
		If we think of a polygon as the floor plan of an art gallery, consider the problem of where to place ``guards'', and how many guards to place, so that every point of the gallery can be seen by some guard.
		A triangulation of a simple polygon is a planar subdivision of (the interior of) P whose vertices are the vertices of P and whose faces are all triangles.
		A triangulation can be viewed as the union of the edges of P and a maximal set of noncrossing diagonals.
	www.cs.wustl.edu /~pless/506/l6.html (1957 words)

	Triangulation
		The simplest way to triangulate them would be to add to the convex hull diagonals from the first point to all of the others.
		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)

	COMPUTATIONAL GEOMETRY PROJECT- ENUMERATION OF ALL POSSIBLE UNQIUE TRIANGULATIONS OF A CONVEX POLYGON
		Triangulations, as we know, are the division of a polygon into non-overlapping triangles by means of joining the diagonals of the polygon in such a way that no two diagonals cross each other.
		Triangulations are part and parcel of the study of Computational geometry and they are needed as a basic step in a large number of algorithms, besides having some truly mind-boggling applications like Shape Skeletonization in Images.
		A triangulation is uniquely characterized by the set of outdegrees of its vertices, and hence such a path always gives us a unique triangulation, giving rise to the bijection.
	www.cs.mcgill.ca /~arajwa/compgeom (2602 words)

	Franklin & Marshall - Polygon triangulation: recursive algorithm
		The recursive algorithm for triangulating the polygon begins by searching for two vertices on opposite sides ot the polygon, and connecting them with a diagonal.
		If the diagonal lies totally within the polygon, the polygon has been decomposed into two smaller polygons; if not, a pair of vertices is found whose diagonal does lie totally within the polygon.
		One of the movies made of the triangulation of a sixteen-sided polygon is accompanied by pauses for questions to the viewer.
	www.fandm.edu /x7741.xml (199 words)

	Triangulating and Extruding Arbitrary Polygons
		There should be 11 polygons in the path, one for each character in the string "Some text" and three for the holes in the letters "o" and "e".
		The GLU library defines a number of callback functions that are invoked during tesselation to resolve intersections of polygon edges and after the tesselation to pass back the resulting triangles to the caller.
		After converting a polygon into a list of triangles with the help of the GLU tesselation functions, the polygon is still flat.
	www.geocities.com /foetsch/extrude/extrude.htm (4560 words)

	PolyTrans-for-XSI, Common Geometry Processing Option Descriptions
		These polygon processing functions allow polygon vertices to be welded together, normals unified, auto-smoothing to be performed, redundant coordinates to be deleted and triangles merged back into n-sided polygons.
		The algorithm allows the number of polygons in the scene to be greatly reduced.
		In those cases when a N-sided polygon must be converted to triangles, up to 3 different triangulation routines can be enabled inside the core Okino software.
	www.okino.com /conv/pt4xsi6.htm (871 words)

	Algorithm Library Design 2003: Project: Polygon Triangulation
		It creates a triangulation of the convex hull of a set of points respecting a set of constrained edges that have to be present in the triangulation, which would be the polygon boundary edges here.
		Afterwards the triangles in the polygon interior have to be selected, i.e., distinguished from those outside of the polygon.
		For polygons embedded in a data structure, such as a triangulation or a polyhedral surface, one wants the result triangles to replace the original polygon and to have the proper neighborhood pointers.
	www.mpi-inf.mpg.de /~kettner/courses/lib_design_03/proj/polygon_triang.html (652 words)

	Triangulation (Site not responding. Last check: 2007-10-29)
		Programs for the Delaunay triangulation of a point set, either in the plane or in higher dimensions, are listed on the pages about convex hull and low-dimensional Voronoi diagrams (since the computations are equivalent!).
		Programs for the triangulation or trapezoidation of a polygon or a set of line segments in the plane are on the polygon page and on the page about constrained Delaunany triangulation.
		In GIS, a triangulation of a function (known as a height field), is known as a TIN.
	www.geom.uiuc.edu /software/cglist/tri.html (224 words)

	Polygon Triangulation - The Code Project - C++ / MFC
		HGRD is a code for triangulation of a polygon with or without holes.
		It can triangulate and also deal with many disconnected polygons treated as a single one (see figure with the triangulation of Norway border line).
		Thanks to this, it was possible to triangulate a complex polygon with around 40000 nodes within 6s (Athlon 1.4GHz DDR 266).
	www.codeproject.com /cpp/hgrd.asp (364 words)

	Franklin & Marshall - Polygon Triangulation: Art Gallery Problem
		We use here a recursive algorithm for triangulating the polygon, which is also shown elsewhere: connect two vertices with a diagonal which lies totally within the polygon; this decomposes the polygon into two smaller polygons.
		If an art gallery can be an arbitrary polygon (each side is a wall of the gallery), then we can guard the gallery if we can guard each triangle formed from a triangulation of the gallery (polygon).
		After triangulating the polygon, we convert the triangulation into its dual: a graph for which each node represents a triangle, and each edge represents a shared side of the triangulation.
	www.fandm.edu /x7740.xml (368 words)

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