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In geometry, a polygon () is a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
s and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes.


Etymology

The word ''polygon'' derives from the Greek adjective πολύς (''polús'') 'much', 'many' and γωνία (''gōnía'') 'corner' or 'angle'. It has been suggested that γόνυ (''gónu'') 'knee' may be the origin of ''gon''.


Classification


Number of sides

Polygons are primarily classified by the number of sides. See the table below.


Convexity and intersection

Polygons may be characterized by their convexity or type of non-convexity: *
Convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...
: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean. * Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. *
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 Johnn ...
: the boundary of the polygon does not cross itself. All convex polygons are simple. * Concave: Non-convex and simple. There is at least one interior angle greater than 180°. *
Star-shaped In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defin ...
: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped. * Self-intersecting: the boundary of the polygon crosses itself. The term ''complex'' is sometimes used in contrast to ''simple'', but this usage risks confusion with the idea of a ''
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 polyg ...
'' as one which exists in the complex Hilbert plane consisting of 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. *
Star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.


Equality and symmetry

* Equiangular: all corner angles are equal. *
Equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
: all edges are of the same length. * Regular: both equilateral and equiangular. *
Cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
: all corners lie on a single
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, called the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. *
Tangential In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
: all sides are tangent to an inscribed circle. * Isogonal or
vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular. * Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential. The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a ''regular
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
''.


Miscellaneous

* Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees. *
Monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
with respect to a given line ''L'': every line
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to L intersects the polygon not more than twice.


Properties and formulas

Euclidean geometry is assumed throughout.


Angles

Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: *
Interior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...
– The sum of the interior angles of a simple ''n''-gon is radians or degrees. This is because any simple ''n''-gon ( having ''n'' sides ) can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular ''n''-gon is \left(1-\tfrac\right)\pi radians or 180-\tfrac degrees. The interior angles of regular
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
s were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular \tfrac-gon (a ''p''-gon with central density ''q''), each interior angle is \tfrac radians or \tfrac degrees. *
Exterior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...
– The exterior angle is the supplementary angle to the interior angle. Tracing around a convex ''n''-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an ''n''-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple ''d'' of 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 aroun ...
and 0° for an angular "eight" or antiparallelogram, where ''d'' is the density or turning number of the polygon. See also orbit (dynamics).


Area

In this section, the vertices of the polygon under consideration are taken to be (x_0, y_0), (x_1, y_1), \ldots, (x_, y_) in order. For convenience in some formulas, the notation will also be used.


Simple polygons

If the polygon is non-self-intersecting (that is,
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 Johnn ...
), the signed
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
is :A = \frac \sum_^( x_i y_ - x_ y_i) \quad \text x_=x_ \text y_n=y_, or, using determinants :16 A^ = \sum_^ \sum_^ \begin Q_ & Q_ \\ Q_ & Q_ \end , where Q_ is the squared distance between (x_i, y_i) and (x_j, y_j). The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive -axis to the positive -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
. This is commonly called the ''
shoelace formula The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian co ...
'' or ''surveyor's formula''. The area ''A'' of a simple polygon can also be computed if the lengths of the sides, ''a''1, ''a''2, ..., ''an'' and the
exterior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...
s, ''θ''1, ''θ''2, ..., ''θn'' are known, from: :\beginA = \frac12 ( a_1 _2 \sin(\theta_1) + a_3 \sin(\theta_1 + \theta_2) + \cdots + a_ \sin(\theta_1 + \theta_2 + \cdots + \theta_)\\ + a_2 _3 \sin(\theta_2) + a_4 \sin(\theta_2 + \theta_3) + \cdots + a_ \sin(\theta_2 + \cdots + \theta_)\\ + \cdots + a_ _ \sin(\theta_)). \end The formula was described by Lopshits in 1963. If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points,
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick i ...
gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter ''p'' and area ''A '', the isoperimetric inequality p^2 > 4\pi A holds. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides ''do'' determine the area. Of all ''n''-gons with given side lengths, the one with the largest area is cyclic. Of all ''n''-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).


Regular polygons

Many specialized formulas apply to the areas of regular polygons. The area of a regular polygon is given in terms of the radius ''r'' of its inscribed circle and its perimeter ''p'' by :A = \tfrac \cdot p \cdot r. This radius is also termed its apothem and is often represented as ''a''. The area of a regular ''n''-gon in terms of the radius ''R'' of its
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
can be expressed trigonometrically as: :A = R^2 \cdot \frac \cdot \sin \frac = R^2 \cdot n \cdot \sin \frac \cdot \cos \frac The area of a regular ''n''-gon inscribed in a unit-radius circle, with side ''s'' and interior angle \alpha, can also be expressed trigonometrically as: :A = \frac\cot \frac = \frac\cot\frac = n \cdot \sin \frac \cdot \cos \frac.


Self-intersecting

The area of a
self-intersecting polygon Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: *the crossed qu ...
can be defined in two different ways, giving different answers: * Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the ''density'' of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure. * Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.


Centroid

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are :C_x = \frac \sum_^ (x_i + x_) (x_i y_ - x_ y_i), :C_y = \frac \sum_^ (y_i + y_) (x_i y_ - x_ y_i). In these formulas, the signed value of area A must be used. For triangles (), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for . The
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any o ...
of the vertex set of a polygon with vertices has the coordinates :c_x=\frac 1n \sum_^x_i, :c_y=\frac 1n \sum_^y_i.


Generalizations

The idea of a polygon has been generalized in various ways. Some of the more important include: * A spherical polygon is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the digon, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
(map making) and in Wythoff's construction of the
uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also f ...
. * A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polytopes are well known examples. * An
apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...
is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions. * A skew apeirogon is an infinite sequence of sides and angles that do not lie in a flat plane. * A
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 polyg ...
is a
configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...
analogous to an ordinary polygon, which exists in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
of two
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
and two imaginary dimensions. * An abstract polygon is an algebraic
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a ''realization'' of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized. * A polyhedron is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called polytopes. (In other conventions, the words ''polyhedron'' and ''polytope'' are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.)


Naming

The word ''polygon'' comes from Late Latin ''polygōnum'' (a noun), from Greek πολύγωνον (''polygōnon/polugōnon''), noun use of neuter of πολύγωνος (''polygōnos/polugōnos'', the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix ''-gon'', e.g. '' pentagon'', ''
dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational s ...
''. The triangle, quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.Mathworld Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example the regular
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
pentagon is also known as the
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 aroun ...
. To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of
quasiregular polyhedra In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the sem ...
, though not all sources use it.


History

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the
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 aroun ...
, a non-convex regular polygon (
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
), appearing as early as the 7th century B.C. on a krater by
Aristophanes Aristophanes (; grc, Ἀριστοφάνης, ; c. 446 – c. 386 BC), son of Philippus, of the deme Kydathenaion ( la, Cydathenaeum), was a comic playwright or comedy-writer of ancient Athens and a poet of Old Attic Comedy. Eleven of his ...
, found at Caere and now in the Capitoline Museum. The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century. In 1952,
Geoffrey Colin Shephard Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in ...
generalized the idea of polygons to the complex plane, where each
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
dimension is accompanied by an imaginary one, to create complex polygons.


In nature

Polygons appear in rock formations, most commonly as the flat facets of crystals, where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of
basalt Basalt (; ) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90% of a ...
, which may be seen at the Giant's Causeway in Northern Ireland, or at the Devil's Postpile in California. In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, the surface of the wax honeycomb made by
bee Bees are winged insects closely related to wasps and ants, known for their roles in pollination and, in the case of the best-known bee species, the western honey bee, for producing honey. Bees are a monophyletic lineage within the superfami ...
s is an array of
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
s, and the sides and base of each cell are also polygons.


Computer graphics

In computer graphics, a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials. Any surface is modelled as a tessellation called
polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex poly ...
. If a square mesh has points (vertices) per side, there are ''n'' squared squares in the mesh, or 2''n'' squared triangles since there are two triangles in a square. There are vertices per triangle. Where ''n'' is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines). The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation. In computer graphics and
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems a ...
, it is often necessary to determine whether a given point P=(x_0,y_0) lies inside a simple polygon given by a sequence of line segments. This is called the point in polygon test.


See also

*
Boolean operations on polygons Boolean operations on polygons are a set of Boolean operations (AND, OR, NOT, XOR, ...) operating on one or more sets of polygons in computer graphics. These sets of operations are widely used in computer graphics, CAD, and in EDA (in integrated ...
*
Complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
*
Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinit ...
*
Cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
* Geometric shape *
Golygon A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a ...
* List of polygons *
Polyform In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle ...
*
Polygon soup A polygon soup is a set of unorganized polygons, typically triangles, before the application of any structuring operation, such as e.g. octree grouping. The term must not to be confused with the "PolySoup" operation available in the 3D package H ...
* Polygon triangulation * Precision polygon * Spirolateral * Synthetic geometry *
Tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling * Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People * Heinrich Sylvester T ...
*
Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...


References


Bibliography

* Coxeter, H.S.M.; ''
Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
'', Methuen and Co., 1948 (3rd Edition, Dover, 1973). * Cromwell, P.; ''Polyhedra'', CUP hbk (1997), pbk. (1999). * Grünbaum, B.; Are your polyhedra the same as my polyhedra? ''Discrete and comput. geom: the Goodman-Pollack festschrift'', ed. Aronov et al. Springer (2003) pp. 461–488.''
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Notes


External links

*

with Greek Numerical Prefixes

with interactive animation
How to draw monochrome orthogonal polygons on screens
by Herbert Glarner
comp.graphics.algorithms Frequently Asked Questions
solutions to mathematical problems computing 2D and 3D polygons

compares capabilities, speed and numerical robustness

Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons {{Authority control Euclidean plane geometry