Polygon

Polygon[{p₁,…,p_n}]

represents a filled polygon with points p_i.

Polygon[{p₁,…,p_n}{{q₁,…,q_m},…}]

represents a polygon with holes {q₁,…,q_m},….

Polygon[{poly₁,poly₂,…}]

represents a collection of polygons poly_i.

Polygon[{p₁,…,p_n},data]

represents a polygon in which coordinates given as integers i in data are taken to be p_i.

Details and Options

Polygon can be used as a geometric region and a graphics primitive.
Polygon[{p₁,…,p_n}] is a plane region, representing all the points inside the closed curve with line segments {p₁,p₂},…,{p_n-1,p_n} and {p_n,p₁}.
A point is an element of the polygon if a ray from the point in any direction in the plane crosses the boundary line segments an odd number of times.

Polygon[{p₁,…,p_n}{{q₁,…,q_m},…}] specifies a polygon with holes consisting of an outer polygon Polygon[{p₁,…,p_n}] and one or several inner polygons Polygon[{q₁,…,q_m}],….
A point is an element of the polygon if it is in the outer polygon but not in any inner polygon.

Polygon[{poly₁,poly₂,…}] is a collection of polygons poly_i with or without holes and is treated as a union of poly_i for geometric computations.

Polygon[{p₁,…,p_n},data] effectively replaces integers i that appear as coordinates in data by the corresponding p_i.

	Polygon[{p₁,…,p_n},{b₁,…,b_n}]	polygon boundary points {p_b₁,…,p_{b_k}}
	Polygon[{p₁,…,p_n},{{o₁,…,o_k}{{i₁,…,i_l},…}]	outer polygon boundary points {p_o₁,…,p_{o_k}} and inner polygon boundary {p_i₁,…,p_{i_l}} etc.
	Polygon[{p₁,…,p_n},{{b₁,…,b_n},{o₁,…,o_k}{{i₁,…,i_l},…},…}]	a collection of several polygons

As a geometric region, the points p_i can have any embedding dimension, but must all lie in a plane and have the same embedding dimension.
In a graphic, the points p_i can be Scaled, Offset, ImageScaled and Dynamic expressions.
Graphics rendering is affected by directives such as FaceForm, EdgeForm, Texture, Specularity, Opacity and color.
FaceForm[front,back] can be used to specify different styles for the front and back of polygons in 3D. The front is defined by the right-hand rule and the direction of the first three points.
The following options and settings can be used in graphics:

VertexColors	Automatic	vertex colors to be interpolated
VertexNormals	Automatic	effective vertex normals for shading
VertexTextureCoordinates	None	coordinates for textures

Point can be used with symbolic points in GeometricScene.

Examples

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Basic Examples (2)

A polygon:

Its graphic image:

Its area:

Differently styled 3D polygons:

Scope (21)

Graphics (11)

Specification (2)

A collection of polygons:

Polygons with multiple vertices:

Styling (6)

Color directives specify the face colors of polygons:

Texture can be used to specify a texture to be used on the faces of polygons:

Texture can work together with different Opacity:

Texture can work together with different Lighting:

FaceForm and EdgeForm can be used to specify the styles of the interiors and boundaries:

In 3D, different properties can be specified for the front and back of faces using FaceForm:

Use FaceForm to set front and back textures differently in 3D:

Colors can be specified at vertices using VertexColors:

Normals can be specified at vertices using VertexNormals for 3D polygons:

Coordinates (3)

Use Scaled coordinates:

Use ImageScaled coordinates in 2D:

Use Offset coordinates in 2D:

Regions (10)

Embedding dimension:

Geometric dimension:

Point membership test:

Get conditions for point membership:

Area:

Centroid:

Distance from a point:

Plot it:

Signed distance from a point:

Plot it:

Nearest point in the region:

Nearest points:

A polygon is bounded:

Get its range:

Integrate over a polygon:

Optimize over a polygon:

Solve equations in a polygon:

Options (7)

VertexColors (2)

Polygon with vertex colors:

Specify vertex colors for 3D polygons:

VertexNormals (1)

Compute normal vectors using the cross product of edge vectors:

A triangle with normals pointing in the direction {1,-1,1}:

Using different normals will affect shading:

VertexTextureCoordinates (4)

Texture-mapped polygon:

Texture mapping with 2D polygons:

Texture mapping with 3D polygons:

Repeat a texture by using non-unified texture coordinate values:

Texture mapping is preceded by VertexColors:

Applications (3)

Define a polygon with vertices:

Regular polygons:

Star polygons:

Define the regular hexagon:

Regular hexagonal tiling:

Get face polygons from PolyhedronData:

Shrink each face with respect to the centroid:

Properties & Relations (4)

GraphicsComplex offers an efficient way to generate a polygon with many shared vertices:

Applying Normal to the graphics complex produces ordinary polygons:

Polygon is a generalization of Triangle:

Polygon is a generalization of Rectangle:

A Simplex with three vertices is a special case of Polygon:

In any number of dimensions starting from 2:

Possible Issues (3)

In 3D, if the vertices are not in a plane, the polygon triangulation can be unpredictable:

Degenerate polygons are not valid geometric regions:

Seams can sometimes appear between individual polygons as a result of antialiasing:

Using a single polygon object avoids any seams:

Neat Examples (3)

Random triangle collections:

Digital petals:

A rotating star:

Introduced in 1988

(1.0)

Updated in 1996

(3.0)

2019

(12.0)

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