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The term complex polygon can mean two different things:

• In geometry, a polygon in the unitary plane, which has two complex dimensions.
• In computer graphics, a polygon whose boundary is not simple.

In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions.^[1]

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

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.^[2]

Self-intersecting polygons are also sometimes included among the complex polygons.^[3] 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

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

• Introduction to Polygons

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