Complex polygon
Polygon in complex space, or which self-intersects
The term complex polygon can mean two different things:
01Geometry
In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions.
A complex number may be represented in the form 
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.
02Computer graphics
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.
Self-intersecting polygons are also sometimes included among the complex polygons. 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".
Sources and credits
This article is adapted from the Wikipedia article “Complex polygon”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Pentagram with vertices.svg by Cmglee, CC BY-SA 4.0
- Regular star polygons.svg by Cmglee, CC BY-SA 3.0
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