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| | Electrical Engineer Fant from Oromocto, likes to spend time fencing, property developers in [http://destinyexpress.net/groups/new-property-launch-singapore/ singapore real estate market] and tea tasting. Was recently visiting Central Sikhote-Alin. |
| The term '''complex polygon''' can mean two different things:
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| *In [[computer graphics]], as a [[polygon]] which is neither [[convex polygon|convex]] nor [[concave polygon|concave]].
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| *In [[geometry]], as a polygon in the [[unitary space|unitary]] plane, which has two [[complex number|complex]] dimensions.
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| ==Computer graphics==
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| [[Image:Complex polygon.svg|160px|right|thumb|A complex (self-intersecting) pentagon]]
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| In the world of computer graphics, a complex polygon is a [[polygon]] which is neither [[convex polygon|convex]] nor concave. This includes any polygon which:
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| *Intersects itself. These include [[star polygon]]s such as the [[pentagram]]:[[Image:Pentagram green.svg|40px|A star polygon]]
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| *Has a boundary comprising discrete circuits, such as a polygon with a hole in it.
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| Therefore, unlike [[simple polygon]]s, a complex polygon may not always be interpreted as a simple polygonal region. Vertices are only counted at the ends of edges, not where edges intersect in space.
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| A formula relating an integral over a bounded region to a closed [[line integral]] may still apply when the "inside-out" parts of the region are counted negatively.
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| Moving around the polygon, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a [[pentagram]] and 0° for an angular "eight".
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| See also: [[orbit (dynamics)]], [[Winding number]].
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| ==Geometry==
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| In [[geometry]], a complex polygon is a polygon in the complex [[Hilbert space|Hilbert]] plane, which has two [[complex number|complex]] dimensions.
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| A [[complex number]] may be represented in the form <math>(a + ib)</math>, where <math>a</math> and <math>b</math> are [[real number]]s, and <math>i</math> is the square root of <math>-1</math>. A complex number lies in a [[complex plane]] having one real and one imaginary dimension, which may be represented as an [[Argand diagram]]. So a single complex dimension is really two dimensions, but of different kinds.
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| The [[unitary space|unitary]] plane comprises two such complex planes, which are [[orthogonal]] to each other. Thus it has two real dimensions <math>x</math> and <math>y,</math> and two imaginary dimensions <math>ix</math> and <math>iy</math>.
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| A complex polygon is a two-dimensional example of the more general [[complex polytope]] in higher dimensions.
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| In a ''real'' plane, a visible figure can be constructed as the ''real conjugate'' of some complex polygon.
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| == References ==
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| * [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]], ''Regular Complex Polytopes'', Cambridge University Press, 1974.
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| == See also ==
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| * [[Simple polygon]]
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| * [[Convex and concave polygons]]
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| * [[Star polygon]]
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| * [[Convex hull]]
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| * [[Nonconvex uniform polyhedron]]
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| * [[Nonzero-rule]]
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| == External links ==
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| * [http://freespace.virgin.net/hugo.elias/graphics/x_polyd.htm Introduction to Polygons]
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| [[Category:Polygons]]
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| {{geometry-stub}}
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Electrical Engineer Fant from Oromocto, likes to spend time fencing, property developers in singapore real estate market and tea tasting. Was recently visiting Central Sikhote-Alin.