# Complex polygon

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The term **complex polygon** can mean two different things:

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

## Computer graphics

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:

- 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.

## Geometry

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 , where and are real numbers, and is the square root of . 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 and and two imaginary dimensions and .

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.

## References

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

## See also

- Simple polygon
- Convex and concave polygons
- Star polygon
- Convex hull
- Nonconvex uniform polyhedron
- Nonzero-rule