Complex polygon

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The term complex polygon can mean two different things:

Contents

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 , where and are real numbers, and is the square root of . Multiples of such as 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

A complex (self-intersecting) pentagon with vertices indicated Pentagram with vertices.svg
A complex (self-intersecting) pentagon with vertices indicated
All regular star polygons (with fractional Schlafli symbols) are complex Regular star polygons.svg
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. [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

Related Research Articles

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In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

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<span class="mw-page-title-main">Regular 4-polytope</span> Four-dimensional analogues of the regular polyhedra in three dimensions

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<span class="mw-page-title-main">3-3 duoprism</span>

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<span class="mw-page-title-main">Regular complex polygon</span> Polygons which have an accompanying imaginary dimension for each real dimension

In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real dimensions, , while a complex polygon exists in two complex dimensions, , which can be given real representations in 4 dimensions, , which then must be projected down to 2 or 3 real dimensions to be visualized. A complex polygon is generalized as a complex polytope in .

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