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Topic: Concave polygon

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Convex polygon

Simple polygon

Star polygon

Regular polygon

Complex polygon

Polygon

Cyclic polygon

List of geometric shapes

Decagon

Constructible polygon

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Triangle

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Perpendicular

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	Concave Polygon Definition - Math Open Reference
		A concave polygon is defined as a polygon with one or more interior angles greater than 180°.
		A concave polygon is the opposite of a convex polygon.
		The area of a concave polygon can be found by treating it as any other irregular polygon.
	www.mathopenref.com /polygonconcave.html (250 words)

	SparkNotes: Polygons: Different Kinds of Polygons
		For a polygon to be convex, all of its interior angles must be less than 180 degrees.
		Another way to think of it is this: the diagonals of a convex polygon will all be in the interior of the polygon, whereas certain diagonals of a concave polygon will lie outside the polygon, on its exterior.
		The center of a regular polygon is the point from which all the vertices of the polygon are equidistant.
	www.sparknotes.com /math/geometry1/polygons/section2.rhtml (247 words)

	Concave - Wikipedia, the free encyclopedia
		Concave lens, a lens with inward-curving (concave) surfaces.
		Concave polygon, a polygon which is not convex.
		Concave function, a type of function which is related to convex functions.
	en.wikipedia.org /wiki/Concave (132 words)

	Convex polygon - Wikipedia, the free encyclopedia
		In geometry, a convex polygon is a simple polygon whose interior is a convex set.
		At least one internal angle of a concave polygon is larger than 180 degrees.
		A concave polygon is often called re-entrant polygon (but in some cases the latter term has a different meaning).
	en.wikipedia.org /wiki/Convex_polygon (180 words)

	Encyclopedia :: encyclopedia : Polygon (Site not responding. Last check: 2007-11-02)
		POLYGON is an Electronic Warfare Tactics Range located on the border between France and Germany.
		A Polygon is a two-dimensional figure, meaning all of the lines in the figure are contained within one plane.
		Regular polygons are polygons in which all sides and angles are congruent.
	www.hallencyclopedia.com /Polygon (254 words)

	Polygons
		Polygons are named according to the number of sides.
		If a polygon has a reflex angle, then it is said to be a concave polygon.
		A regular polygon's sides are all of the same length and its angles are the same size.
	www.mathsteacher.com.au /year7/ch09_polygons/05_polygon/pol.htm (225 words)

	Drawing Polygons, efficient correct polygon rendering (Site not responding. Last check: 2007-11-02)
		A polygon is a bounded region of a plane.
		You can split the edge pixels between the polygons by figuring out how much of the pixel is in each polygon and computing a weighted average of the color of the polygons at each pixel and color the edge pixels with the this average color.
		The inputs to poly() are a color "c" to fill the polygon with, "n" the number of vertices, and a vector of points that represent the polygon vertices.
	gameprogrammer.com /5-poly.html (4272 words)

	Concave - Search Results - MSN Encarta
		Concave, shape of a surface curving inward, or away from the eye.
		Every interior angle of a convex polygon is less than 180°, while at least one angle of a concave polygon is...
		Lenses made with surfaces of small radii have the shorter focal lengths.
	ca.encarta.msn.com /Concave.html (107 words)

	math lessons - Concavity
		In mathematical analysis, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions.
		Similarly, a differentiable function f is concave on an interval if its derivative function f ′ is decreasing on that interval: a concave function has a decreasing slope.
		Equivalently, f(x) is concave on [a, b] iff the function −f(x) is convex on every subinterval of [a, b].
	www.mathdaily.com /lessons/Concavity (394 words)

	Drawing Polygons, efficient correct polygon rendering (Site not responding. Last check: 2007-11-02)
		A polygon is a bounded region of a plane.
		Concave polygons have sides that are "caved" in.
		The inputs to poly() are a color "c" to fill the polygon with, "n" the number of vertices, and a vector of points that represent the polygon vertices.
	www.gameprogrammer.com /5-poly.html (4272 words)

	Gamasutra - Features - "Crashing into the New Year" [02.10.00]
		Our first step is to create perpendicular vectors for each of the polygon edges and a vector from the test point to the first vertex of each edge.
		One method for determining if the test point is inside the concave polygon comes from the idea that a circle is 360 degrees.
		A better strategy is to divide the polygon into quadrants centered on the test point, as in Figure 4.
	www.gamasutra.com /features/20000210/lander_01.htm (744 words)

	Glossary For Ear Cutting
		The convex hull CH(P) of a polygon P is the smallest convex polygon that contains P.
		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 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.
	www-cgrl.cs.mcgill.ca /~godfried/teaching/cg-projects/97/Ian/glossary.html (468 words)

	Polygon types (Site not responding. Last check: 2007-11-02)
		For example, an algorithm for splitting a polygon into a number of 3 vertex facets is trivial for convex polygons and quite problematic for polygons with holes.
		Rectangular or 4 vertex polygons are generated from gridded datasets and polygonal approximations of 2D surfaces.
		Such polygons can be turned into concave polygons by introducing 2 coincident edges, between the solid and hole polygons.
	astronomy.swin.edu.au /~pbourke/modelling/polygon (351 words)

	New Page 0 (Site not responding. Last check: 2007-11-02)
		Regular Polygons have sides that are all the same length and angles that are all equal.
		Polygons that are not regular have some sides of different lengths and/or angles of different degrees.
		A polygon is concave if a line segment can be drawn between two points somewhere inside the polygon also passes outside the polygon.
	www.pelhamweb.com /pes/MrRobertson/polygons.htm (571 words)

	Lesson 5.01 Polygons
		A test to determine if a polygon is concave, is to extend all the sides.
		If the sides extend into the interior of the polygon, it is concave.
		The sum of the measures of the angles of a convex polygon with n sides is 180 (n - 2).
	www.flvs.net /_students/showcase_flvs/math/geometry/Module5/LesMod5/5_01.htm (239 words)

	Weiler-Atherton Algorithm
		The Weiler-Atherton algorithm is capable of clipping a concave polygon with interior holes to the boundaries of another concave polygon, also with interior holes.
		The exterior boundaries of the polygons are described clockwise, and the interior boundaries or holes are described counter-clockwise.
		Polygons outside the CP are found using the same procedure, except that the initial intersection vertex is obtained from the leaving list and the CP vertex list is followed in the reverse direction.
	www.anirudh.net /practical_training/main/node10.html (577 words)

	Dummies::Sizing Up the Area of a Polygon
		Not only can polygons be classified by the number of sides they have and by their angles, but they can also be grouped according to some of their qualities.
		When two different radii in a polygon are drawn to two consecutive vertices, a central angle is formed in the center of the polygon (see Figure 1).
		Theorem 5-11: The measure of a central angle in a regular polygon is equal to 360° divided by the number of sides of the polygon.
	www.dummies.com /WileyCDA/DummiesArticle/id-1195.html (803 words)

	MATHEMATICS DICTIONARY (Site not responding. Last check: 2007-11-02)
		A polygon of 3 sides is a triangle; of 4 sides, a quadrilateral; of 5 sides, a pentagon; of 6 sides, a hexagon; of 7 sides, a heptagon; of 8 sides, an octagon; of 9 sides, a nonagon; of 10 sides, a decagon; of 12 sides, a dodecagon; of «-sides, an n-gon.
		A polygon is concave if and only if there is a straight line which passes through the interior of the polygon and cuts the polygon in four or more points.
		A concave polygon has an interior if no side touches any other side, except at a vertex, and no two vertices coincide (i.e., if it is a simple closed curve or a Jordan curve).
	lines-and-dots.org /polygon.html (335 words)

	Geometry Definitions
		Concave polygons look like they are collapsed or have one or more angles dented in.
		A face is a polygon by which a solid object is bound.
		A regular polygon has sides that are all the same length and angles that are all the same size.
	www.learner.org /channel/courses/learningmath/geometry/keyterms.html (2247 words)

	Convex and Concave Polygons (Site not responding. Last check: 2007-11-02)
		Convex and concave polygons are mutually exclusive polygon types that span the whole set of polygons.
		A polygon is defined to be convex if for any two points that lie within the polygon, the line segment connecting them is also inside the polygon.
		Thus, the polygons are classified as concave polygons.
	www-static.cc.gatech.edu /gvu/multimedia/nsfmmedia/graphics/elabor/polyscan/polytypes.html (118 words)

	NSDL Metadata Record -- Concave Polygon -- from MathWorld
		A concave polygon is a polygon that is not convex.
		A polygon is concave iff at least one of its internal angles is greater than 180?.
		A concave polygon must have at least four sides.
	nsdl.org /mr/696431 (58 words)

	CS184 : Discussion Section 3 - Scan Conversion Distillation
		Triangles are the only polygons which are guaranteed to be convex and planar, and we'll say why this is important later in the course.
		Concave polygons don't have the nice properties of convex polygons, and are thus more complicated.
		If it's possible to pull the polygon away (imagine that the polygon's edges are traced in string on a table) then the point is outside, otherwise it's inside.
	www.cs.berkeley.edu /~ddgarcia/cs184/r3 (1484 words)

	Polygon Edit > Boolean > Subtract, Intersect, Union
		Technically, a polygon is concave if you can connect any two vertices with an edge that runs outside of the polygon.
		However, if you are combining polygons that are extremely large or small, you may need to change it.
		The polygon Boolean tools let you combine two closed polysets in different ways, similar to the Boolean operators for shells (see the NURBS Modeling Book).
	www.alias.com /eng/support/studiotools/documentation/Reference/PolygonEditBooleanSubtract.html (577 words)

	Clockwise or counterclockwise polygon in a plane (Site not responding. Last check: 2007-11-02)
		As a consequence the test can also be used to determine whether or not a polygon is concave or convex.
		For the more general case where the polygons may be convex, it is necessary to consider the sign of the cross product between adjacent edges as one moves around the polygon.
		For a convex polygon all the cross products of adjacent edges will be the same sign, a concave polygon will have a mixture of cross product signs.
	debian.fmi.uni-sofia.bg /~sergei/cgsr/docs/clockwise.htm (373 words)

	Basic Math FAQ: Answers (Site not responding. Last check: 2007-11-02)
		A polygon is a many-sided planar figure composed of verticies and edges.
		A polygon is convex if for any two points P1,P2 inside the polygon, all points on the line segment which connects P1 and P2 are inside the polygon.A concave polygon is one which is not complex.
		A polyhedron is a solid that is bounded by a set of polygons whose edges are each a member of an even number of polygons (and that satisfies some additional constraints).
	www-static.cc.gatech.edu /gvu/multimedia/nsfmmedia/graphics/elabor/math/mathfaq_polys.html (130 words)

	Determining Whether A Point Is Inside A Complex Polygon
		Figure 1 demonstrates a typical case of a severely concave polygon with 14 sides.
		The solution is to compare each side of the polygon to the Y (vertical) coordinate of the test point, and compile a list of nodes, where each node is a point where one side crosses the Y threshold of the test point.
		So, the test point is outside the polygon, as indicated by the even number of nodes (two and two) on either side of it.
	www.alienryderflex.com /polygon (851 words)

	Chapter 11 - OpenGL Programming Guide
		Figure 11-1 shows some contours of polygons that require tessellation: from left to right, a concave polygon, a polygon with a hole, and a self-intersecting polygon.
		As a complex polygon is being described and tessellated, it has associated data, such as the vertices, edges, and callback functions.
		The first polygon consists of two contours; the outer one is wound counterclockwise, and the "hole" is wound clockwise.
	www.glprogramming.com /red/chapter11.html (4827 words)

	Math Thematics Book 2 (Site not responding. Last check: 2007-11-02)
		If a polygon is convex, every line that contains two of its vertices passes through the interior of the polygon.
		For a concave polygon, there is at least one line containing two vertices that does not pass through the interior of the polygon.
		In a regular polygon, all the segments are the same length and all the angles are equal in measure.
	www.kent.k12.wa.us /staff/DavidChesley/two6.1.htm (215 words)

	Convex and Concave Polygons (Site not responding. Last check: 2007-11-02)
		Convex and concave polygons are mutually exclusive polygon types that span the whole set of polygons.
		A polygon is defined to be convex if for any two points that lie within the polygon, the line segment connecting them is also inside the polygon.
		Thus, the polygons are classified as concave polygons.
	www.cc.gatech.edu /gvu/multimedia/nsfmmedia/graphics/elabor/polyscan/polytypes.html (118 words)

	Concave polygon splitting - GameDev.Net Discussion Forums
		Posted - 11/28/2003 5:03:28 AM I am determining which polygons in the mesh are in each side of the plane and putting them in a front and back array then determining which cross the plane, finding the point of intersection and creating two new polygons from the one original.
		I am determining which polygons in the mesh are in each side of the plane and putting them in a front and back array then determining which cross the plane, finding the point of intersection and creating two new polygons from the one original.
		Concave polygons may or may not have all vertices in a single plane (rounding errors and the like).
	www.gamedev.net /community/forums/ViewReply.asp?id=1215137 (1319 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		Date: Fri, 1 Nov 96 04:37:14 UT From: [Permission pending] To: rusin@washington.math.niu.edu Subject: Re: [Q] subdividing concave polygons into convex Dear Dave, I am desperately looking for an efficient algorithm to divide a concave polygon to a set of triangles.
		I have no problems to understand your method in finding concave vertices, however, I seem to have a little bit difficulty in following your method to divide the concave polygons.
		The proplem is, however, that the above mentioned triangle does not necessarily lie in the polygon.
	www.math.niu.edu /~rusin/known-math/96/concave (639 words)

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