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The term complex polygon can mean two different things:

In computer graphics, a polygon which is neither convex nor concave.
In geometry, a polygon in the unitary plane, which has two complex dimensions.

Computer graphics
A complex (self-intersecting) pentagon

In the world of computer graphics, a complex polygon is a polygon which is neither convex nor concave. This includes any polygon which:

intersects itself. These include star polygons such as the pentagram:A star polygon

has a boundary comprising discrete circuits, such as a polygon with a hole in it.

Therefore, unlike simple polygons, 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.

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".

See also: orbit (dynamics), Winding number.

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 (a + ib), where a and b are real numbers, and i is the square root of -1. 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.

The unitary plane comprises two such complex planes, which are orthogonal to each other. Thus it has two real dimensions x and y, and two imaginary dimensions ix and iy.

A complex polygon is a two-dimensional example of the more general complex polytope in higher dimensions.

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

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

See also

Simple polygon
Star polygon
Convex hull
Nonconvex uniform polyhedron

External links

Introduction to Polygons

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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