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

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a polygon in the

unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...

plane, which has two

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

dimensions. * In

computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

, a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

whose boundary is not

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

Geometry

, a complex polygon is a polygon in the complex

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...

plane, which has two

dimensions. A

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

may be represented in the form

(a + ib)

, where

a

and

b

are

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, and

i

is the square root of

-1

. Multiples of

i

such as

ib

are called ''

imaginary number An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...

s''. A complex number lies in a

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

having one real and one imaginary dimension, which may be represented as an

Argand diagram In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary ...

. So a single complex dimension comprises two spatial dimensions, but of different kinds - one real and the other imaginary. The

plane comprises two such complex planes, which are

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

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

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

In computer graphics, a complex polygon is a

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.Paul Bourke
Polygons and meshes:Surface (polygonal) Simplification
1997. (retrieved May 2016) 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 In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...

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 A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...

and 0° for an angular "eight".

References

Citations

Bibliography

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

External links