complex polygon

{{See|Complex polytope#Regular complex polygons}}

In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions.Coxeter, 1974.

A complex number may be represented in the form $(a\; +\; ib)$, where $a$ and $b$ are real numbers, and $i$ is the square root of $-1$. Multiples of $i$ such as $ib$ are called imaginary numbers. 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 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.

{{See also|orbit (dynamics)|winding number}}

File:pentagram_with_vertices.svg

File:regular_star_polygons.svgs (with fractional Schläfli symbols) 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.Rae Earnshaw, Brian Wyvill (Ed); New Advances in Computer Graphics: Proceedings of CG International ’89, Springer, 2012, page 654.

Self-intersecting polygons are also sometimes included among the complex polygons.Paul Bourke; [http://paulbourke.net/geometry/polygonmesh/ Polygons and meshes:Surface (polygonal) Simplification] 1997. (retrieved May 2016) 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 angular "eight".

• Regular polygon
• Convex hull
• Nonzero-rule
• List of self-intersecting polygons

{{reflist}}

• Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974.

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

Category:Types of polygons

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