Complex polygon

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.

Geometry

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 $(a + ib)$, 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.

Computer graphics

A complex (self-intersecting) pentagon

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".

See also: orbit (dynamics), Winding number.

See also

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

References

Citations

1. Coxeter, 1974.
2. Rae Earnshaw, Brian Wyvill (Ed); New Advances in Computer Graphics: Proceedings of CG International ’89, Springer, 2012, Page 654.
3. Paul Bourke; Polygons and meshes:Surface (polygonal) Simplification 1997. (retrieved May 2016)

Bibliography

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

External links

• Introduction to Polygons