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{{Short description|Polygon in complex space, or which self-intersects}}
{{Refimprove|date=October 2009}}
{{unfocused |date=May 2024}}
The term '''''complex polygon''''' can mean two different things:

* In [[geometry]], a polygon in the [[unitary space|unitary]] plane, which has two [[complex number|complex]] dimensions.
* In [[computer graphics]], a [[polygon]] whose boundary is not [[Simple polygon|simple]].

==Geometry==
{{See|Complex polytope#Regular complex polygons}}
In [[geometry]], a complex polygon is a polygon in the complex [[Hilbert space|Hilbert]] plane, which has two [[complex number|complex]] dimensions.<ref>Coxeter, 1974.</ref>

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>. Multiples of <math>i</math> such as <math>ib</math> are called ''[[imaginary number]]s''. 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 comprises two spatial dimensions, but of different kinds - one real and the other imaginary.

The [[unitary space|unitary]] plane comprises two such complex planes, which are [[orthogonal]] to each other. Thus it has two real dimensions and two imaginary dimensions.

A '''complex polygon''' is a (complex) two-dimensional (i.e. four spatial dimensions) analogue of a real polygon. As such it is an example of the more general [[complex polytope]] in any number of complex dimensions.

In a ''real'' plane, a visible figure can be constructed as the ''real conjugate'' of some complex polygon.

== Computer graphics ==
{{See also|orbit (dynamics)|winding number}}
[[File:pentagram_with_vertices.svg|thumb|A complex (self-intersecting) pentagon with vertices indicated]]
[[File:regular_star_polygons.svg|thumb|All regular [[star polygon]]s (with fractional [[Schläfli symbol]]s) are complex]]

In computer graphics, a complex polygon is a [[polygon]] which has a boundary comprising discrete circuits, such as a polygon with a hole in it.<ref>Rae Earnshaw, Brian Wyvill (Ed); New Advances in Computer Graphics: Proceedings of CG International ’89, Springer, 2012, page 654.</ref>

Self-intersecting polygons are also sometimes included among the complex polygons.<ref>Paul Bourke; [http://paulbourke.net/geometry/polygonmesh/ Polygons and meshes:Surface (polygonal) Simplification] 1997. (retrieved May 2016)</ref> Vertices are only counted at the ends of edges, not where edges intersect in space.

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.

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 [[crossed rectangle|angular "eight"]].

== See also ==
* [[Regular polygon]]
* [[Convex hull]]
* [[Nonzero-rule]]
* [[List of self-intersecting polygons]]

== References ==

=== Citations ===
{{reflist}}

=== Bibliography ===
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]], ''Regular Complex Polytopes'', Cambridge University Press, 1974.

== External links ==
* [https://web.archive.org/web/20060923023349/http://freespace.virgin.net/hugo.elias/graphics/x_polyd.htm Introduction to Polygons]

[[Category:Types of polygons]]

{{geometry-stub}}

Complex polygon - Revision history

imported>Jlwoodwa: tag as unfocused