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 plane, which has two
complex 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 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
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 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
complex 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
, where
and
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
is the square root of
. Multiples of
such as
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 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
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 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
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.
[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 and 0° for an
angular "eight".
See also
*
Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
*
Convex hull
*
Nonzero-rule
In two-dimensional computer graphics, the non-zero winding rule is a means of determining whether a given Point (geometry)#Points in Euclidean geometry, point falls within an enclosed curve. Unlike the similar even-odd rule, it relies on knowing t ...
*
List of self-intersecting polygons
References
Citations
Bibliography
*
Coxeter, H. S. M., ''Regular Complex Polytopes'', Cambridge University Press, 1974.
External links
Introduction to Polygons
Types of polygons
{{geometry-stub