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Complex polygon

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(Redirected from Nonconvex polygon) Polygon in complex space, or which self-intersects
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The term complex polygon can mean two different things:

Geometry

Further information: Complex polytope § Regular complex polygons

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 + i b ) {\displaystyle (a+ib)} , where a {\displaystyle a} and b {\displaystyle b} are real numbers, and i {\displaystyle i} is the square root of 1 {\displaystyle -1} . Multiples of i {\displaystyle i} such as i b {\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.

Computer graphics

See also: orbit (dynamics) and winding number
A complex (self-intersecting) pentagon with vertices indicated
All regular star polygons (with fractional Schläfli symbols) are complex

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".

See also

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

External links


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