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Topic: Simple polygon

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	Polygons ( POLYGON )
		A polygon is called simple if all nodes of the graph induced by its segments have degree two and it is called weakly simple, if its segments are disjoint except for common endpoints and if the chain does not cross itself.
		When a weakly simple polygon P is traversed either the bounded region is consistently to the left of P or the unbounded region is consistently to the left of P.
		P is initialized to the polygon with vertex sequence pl.
	www.engin.umich.edu /class/me558/manuals/LEDA/POLYGON.html (791 words)

	An Efficient Adaptive Algorithm for Constructing the Convex Differences Tree of a Simple Polygon (Site not responding. Last check: )
		The convex differences tree (CDT) representation of a simple polygon is useful in computer graphics, computer vision, computer aided design and robotics.
		The root of the tree contains the convex hull of the polygon and there is a child node recursively representing every connectivity component of the set difference between the convex hull and the polygon.
		For simply shaped polygons, where K is a constant, the algorithm is linear.
	www.cs.huji.ac.il /~arir/acdt-abs.html (185 words)

	VizUp, polygon reduction software tool, polygon reducer for optimize 3D models file size - Home
		These models may consist of many hundreds of thousands of polygons and be practically impossible to process in real-time visualization.
		VizUp is an ad hoc polygon reduction system that enables you to reduce the number of polygons in a complex 3D model while retaining the quality and appearance of the original.
		With VizUp in place, the process of polygon reduction is downright simple and does not require any advanced knowledge of CAD or CAE.
	www.vizup.com (247 words)

	Pipalia Village \| Rājkot Map - hic sunt polygona
		Then editing is enabled only when several users agree that the mapygon needs to be updated.
		The term mapygon denotes representation of real object on the map using the shape of a simple polygon.
		Mapygons are sorted by geographical location into one- to five-level structure.
	www.mapygon.com /india/gujarat/rajkot/rajkot/pipalia-village (339 words)

	[Exaflop.org] Simple Polygon Rendering
		The polygon rendering techniques I am going to describe are very easy to code and understand but, are certainly not the fastest (they were my first - hence the sloppy code).
		There are many ways to render a polygon but, by far the easiest is to trace along the edges of the polygon storing the leftmost and rightmost x coorinate for each row of the polygon then filling horizontly between these points.
		This is simple to implement as in the case of deltax > deltay we have a section of code which is entered only when we move a row....
	www.exaflop.org /docs/naifgfx/naifpoly.html (3390 words)

	Glossary For Ear Cutting
		Given a triangulated simple polygon, the dual-tree is the graph generated by plotting a vertex at each triangle and edges joining vertices in adjacent triangles (triangles which share a diagonal).
		A good sub-polygon of a simple polygon P, denoted by GSP, is a sub-polygon whose boundary differs from that of P by at most one edge.
		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.
	cgm.cs.mcgill.ca /~godfried/teaching/cg-projects/97/Ian/glossary.html (468 words)

	Simple polygon - Wikipedia, the free encyclopedia
		A polygon that is not simple is a complex polygon, and does not always have a well-defined inside and outside.
		A simple polygon is topologically equivalent to a disk.
		Although convex polygons are easy to triangulate, triangulating a general simple polygon is more difficult because we have to avoid adding edges that cross outside the polygon.
	en.wikipedia.org /wiki/Simple_polygon (260 words)

	Polygon - Wikipedia, the free encyclopedia
		If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
		The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns).
		Equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p.
	en.wikipedia.org /wiki/Polygon (973 words)

	Polygon - Biocrawler (Site not responding. Last check: )
		A polygon (literally "many angle", see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments.
		A simple polygon is called convex if it has no internal angles greater than 180° otherwise it is called concave.
		The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n).
	www.biocrawler.com /encyclopedia/Polygon (804 words)

	[No title]
		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.
		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/506/l6.html (1957 words)

	Simple Polygonizations (Erik Demaine)
		A simple polygonization of a set of points is a simple polygon whose vertices are precisely that set of points.
		Simple polygonizations are also called simple polygonalizations, or planar tours, or planar traveling salesman (TSP) tours.
		Several of these upper-bound papers are not directly about the polygonization problem, but rather are about one of two other problems that have been shown to be related in the sense that a bound on the related problem induces a bound on the polygonization problem.
	theory.lcs.mit.edu /~edemaine/polygonization (1248 words)

	[No title] (Site not responding. Last check: )
		A principal vertex pi of a simple polygon P is called an ear if the diagonal (p[i-1], p[i+1]) that bridges p[i] lies entirely in P. We say that two ears p[i] and p[j] are non-overlapping if the interior of triangle (p[i-1], p[i], p[i+1]) does not intersect the interior of triangle (p[j-1], p[j], p[j+1]).
		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 first simple polygon triangulation "algorithm" was proposed by Lennes in 1911 via recursive diagonal insertion and runs in O (n^2) time (why was Lennes considering the triangulation problem in 1911?).
	www.mema.ucl.ac.be /~wu/FSA2716-2002/project.html (1423 words)

	Polygon Puzzler (Site not responding. Last check: )
		We define a simple polygon as an area enclosed by endpoint-connected line segments such that no line segment intersects another (except for adjoining segments at their endpoints).
		A planar polygon is a polygon whose vertices all lie in the same plane.
		The output should be the area of the polygons specified by the input and should be rounded to the nearest 1/1000 (i.e., three places after the decimal point should be printed).
	acm.uva.es /p/v5/578.html (275 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].
		These polygons are computed from the trapezoidal decomposition by checking whether the two vertices of the original polygon lie on the same side.
		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)

	Polygons ( POLYGON )
		A polygon is called simple if all nodes of the graph induced by its segments have degree two and it is called weakly simple, if its segments are disjoint except for common endpoints and if the chain does not cross itself.
		When a weakly simple polygon P is traversed either the bounded region is consistently to the left of P or the unbounded region is consistently to the left of P.
		P is initialized to the polygon with vertex sequence pl.
	www.algorithmic-solutions.info /leda_manual/POLYGON.html (1073 words)

	Straight Skeleton of a Simple Polygon
		The straight skeleton of a simple polygon is defined by shrinking the polygon by translating each of its edges at a fixed rate, keeping sharp corners at the reflex vertices, and watching where the vertices go.
		However, the straight skeleton of a nonconvex polygon has fewer edges than the medial axis, and all its edges are line segments; the medial axis also has parabolic arcs around each reflex vertex.
		There are already near-linear-time algorithms for two special cases: c-oriented polygons (where each edge is parallel to one of constant number of lines) [5] and polygons with very few reflex vertices.
	compgeom.cs.uiuc.edu /~jeffe/open/skeleton.html (487 words)

	The Two-Ears Theorem
		The simple polygon is a quadrilateral and has two ears, E1 and E2, as shown below.
		Either polygon P has an ear at vertex pi or it does not (i.e.
		Polygon P does not have an ear at p
	www.personal.kent.edu /~rmuhamma/Compgeometry/MyCG/TwoEar/two-ear.htm (1265 words)

	Geometry Lab: Triangulation of a simple polygon
		One can describe the problem as follows: If a polygon is given that has n edges, then the n diagonals are searched, which divide the polygon into n-2 triangles.
		The y-monotonous polygon, developed by the diagonals, can be divided into triangles in linear time with a Greedy algorithm.
		As soon as the polygon is closed, the triangulation is computed and can be displayed by clicking the checkmarks at the lower edge of the screen.
	web.informatik.uni-bonn.de /I/GeomLab/Triangulation/index.html.en (235 words)

	Intersections for a 2D Set of Segments
		Note that the resulting simple polygons may not be disjoint since one could be contained inside another.
		In fact, the decomposition inclusion hierarchy is based on the inclusion winding number of each simple polygon in the original non-simple one (see Algorithm 3 about Winding Number Inclusion).
		However, when two polygons overlap, the sweep line strategy of the Bentley-Ottmann algorithm can be adapted to perform a set operation on any two simple polygons.
	geometryalgorithms.com /Archive/algorithm_0108/algorithm_0108.htm (3391 words)

	Convex Hull of a 2D Simple Polyline
		However, this month's algorithm uses the given sequential ordering of a simple polygon's edges along with a similar algorithm using a "deque" (a double-ended queue).
		Initially, the O(n) improvement for simple polygons was proposed, and an algorithm implementation given, by [Sklansky, 1972].
		All simple polygon or polyline convex hull algorithms implement this strategy in one form or another.
	softsurfer.com /Archive/algorithm_0203/algorithm_0203.htm (2176 words)

	Straight Skeleton of a Simple Polygon
		The straight skeleton of a simple polygon is defined by shrinking the polygon by translating each of its edges at a fixed rate, keeping sharp corners at the reflex vertices, and watching where the vertices go.
		However, the straight skeleton of a nonconvex polygon has fewer edges than the medial axis, and all its edges are line segments; the medial axis also has parabolic arcs around each reflex vertex.
		Another special case that seems likely to have a fast solution is polygons whose smallest (external) reflex angle is at least 90 degrees, or even just bounded away from zero.
	granmapa.cs.uiuc.edu /~jeffe/open/skeleton.html (487 words)

	LF vol. 2 OpenGL Programming: Simple Polygon Rendering (Site not responding. Last check: )
		The first example is a simple OpenGL program that draws a number of orbits in a chaotic map (The standard map).
		Using polygon drawing try attaching boxes, diamonds, or whatever to the end of the pendulum.
		In the next issue (March 1998) we will continue to explore polygons, modeling and cover in more detail some of the commands your are already familiar with.
	redhat.com /mirrors/LDP/linuxfocus/Portugues/January1998/article17.html (1939 words)

	Polygon Reduction - by Stan Melax (Site not responding. Last check: )
		Look at the Polygon Reduction article from the November '98 issue of Game Developer Magazine which explains how this polygon reduction technique works.
		A Polygon Reduction Plugin for 3DS Max (with support for texture coordinates).
		Polygon Reduction Plugin for Lightwave (by Kasper J. Wessing).
	www.melax.com /polychop/index.html (202 words)

	polygon
		a cyclic polygon is a polygon where the sides are chords of a given circle
		A so-called control polygon is used to define a curve by points near the curve, not by points on the curve.
		In the field of Roman ancient history the so called Thijssen polygons are in use: these polygons have been constructed around Roman district capitals, to approximate the districts' boundaries.
	www.2dcurves.com /line/linep.html (340 words)

	Chvatal's Art Gallery Theorem
		Show that for such a diagonal triangulation of the polygon, its vertices can be colored with three colors, such that all three colors are present in every triangle of the triangulation.
		Observe that, in a diagonal triangulation, the sides of any triangle are either sides or the diagonals of the polygon, and at most 2 may be of the latter kind.
		However, it is based on the premise that any simple polygon has at least one diagonal that lies entirely in the interior of the polygon.
	www.cut-the-knot.org /Curriculum/Combinatorics/Chvatal.shtml (875 words)

	lf17, SoftwareDevelopment: OpenGL Programming: Simple Polygon Rendering
		The first example is a simple OpenGL program that draws a number of orbits in a chaotic map (The standard map).
		Using polygon drawing try attaching boxes, diamonds, or whatever to the end of the pendulum.
		In the next issue (March 1998) we will continue to explore polygons, modeling and cover in more detail some of the commands your are already familiar with.
	www.linuxfocus.org /English/January1998/article17.html (1978 words)

	CS180 Project 3 -- Simple Polygon Detection Program (Site not responding. Last check: )
		You are required to write a program that will determine the names of some basic convex polygons (closed figures) given the details about the number of sides it is bounded by and the side lengths in some cases.
		If the polygon is determined to a triangle, then the user has to consider the lengths of the sides to determine more specific details.
		Assume that the user always enters a valid entry for the number of sides of the polygon.
	www.cs.purdue.edu /homes/bxd/180/proj3.print.html (204 words)

	Maximizing the Area of an Axis-Symmetric Polygon Inscribed by a Simple Polygon
		We propose an algorithm for solving this problem, analyze its complexity, and describe our implementation of it (for the case of a convex polygon).
		The algorithm is based on building and investigating a planar map, each cell of which corresponds to a different configuration of the inscribed polygon.
		Maximizing the Area of an Axis-Symmetric Polygon Inscribed by a Simple Polygon [pdf]
	www.cs.tau.ac.il /CGAL/Projects/Max_inscribed_poly.php (154 words)

	On-Line Search in a Simple Polygon - Kleinberg (ResearchIndex) (Site not responding. Last check: )
		Abstract: We consider a number of search and exploration problems, from the perspective of robot navigation in a simple polygon.
		These problems are "on-line" in the sense that the robot does not have access to the map of the polygon; it must make decisions as it proceeds, based only on what it has seen so far.
		For the problem of exploring a simple rectilinear polygon (under the L 1 norm), Deng, Kameda, and Papadimitriou give a 2-competitive deterministic algorithm; we present a randomized exploration...
	citeseer.ist.psu.edu /200352.html (420 words)

	Problem 56: Packing Unit Squares in a Simple Polygon
		What is the complexity of deciding whether a given number of axis-parallel unit squares can be packed into a simple polygon (without holes)?
		The problem is known to be NP-complete for polygons with holes [
		FPT81], even if the polygon is an orthogonal polygon with all coordinates being multiples of 1/2.
	maven.smith.edu /~orourke/TOPP/P56.html (164 words)

	Non Simple Polygon Filling from Joe Mihalich on 2002-11-07 (www-svg@w3.org from November 2002)
		Hi, I'm having a problem with an overlapping polygon.
		Suppose I draw a polygon shaped like a star, where the lines Overlap, causing a hole in the middle of the star.
		However, the SVG representation Of that same polygon causes viewers to not clip around the Hole and the entire inner part of the star gets filled.
	lists.w3.org /Archives/Public/www-svg/2002Nov/0019.html (185 words)

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