CrossingPolygon

CrossingPolygon[{p1,p2,…,pn}]

gives a Polygon representing all points for which a ray from the point in any direction in the plane crosses the line segments {p1,p2},…,{pn-1,pn},{pn,p1} an odd number of times.

CrossingPolygon[{{p11,p12,…},{p21,p22,…},…}]

gives a Polygon from the line segments {p11,p12},…,{p21,p22},….

Details and Options

  • CrossingPolygon is also known as even–odd filling rule.
  • A point is in CrossingPolygon if a ray starting at that point to infinity in any direction will cross the boundary curves an odd number of times. The number of ray crossings is given by CrossingCount.
  • The number of ray crossings is given below for each region.
  • CrossingPolygon is used to define a polygon from possibly self-intersecting closed curves.
  • CrossingPolygon[{p1,p2,…,pn}] is effectively equivalent to Polygon[{p1,p2,…,pn}].
  • CrossingPolygon[{{p11,p12,…},{p21,p22,…},…}] is, in general, different than Polygon[{{p11,p12,…},{p21,p22,…},…}] since the former will use the ray crossing rule for all closed curves {pi1,pi2,…}. The latter is the union of polygons Polygon[{pi1,pi2,…}].
  • The points pi can have any length but must all lie in a plane.
  • CrossingPolygon takes the same options as Polygon.

    • List of all options

Examples

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Basic Examples  (2)

Define a polygon:

Construct a polygon from a self-intersecting contour:

Compute its area:

Scope  (11)

Basic Uses  (5)

Define a 2-dimensional polygon:

Three-dimensional polygons:

-dimensional polygons:

Construct polygons from self-intersecting contours:

Multiple contours:

Self-Intersecting Contours  (3)

CrossingPolygon works on self-intersecting contours:

Overlapping contour segments:

Multiple contours:

Multiple Contours  (3)

CrossingPolygon works on multiple contours:

Intersecting contours:

Self-intersecting contours:

Options  (6)

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  (3)

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)

Basic Applications  (1)

Polygon construction from contours:

Computer Graphics  (1)

Turtle drawing polygon. Advance 20 steps, always turning 110° toward the left:

Computational Geometry  (1)

Generate random polygons:

Properties & Relations  (3)

CrossingPolygon is effectively equivalent to Polygon for a single contour:

CrossingPolygon is, in general, different than Polygon for multiple intersecting contours:

WindingPolygon is an alternate polygon constructor:

Possible Issues  (1)

The points in CrossingPolygon must all lie on a plane:

Neat Examples  (1)

Digital petals:

Wolfram Research (2019), CrossingPolygon, Wolfram Language function, https://reference.wolfram.com/language/ref/CrossingPolygon.html.

Text

Wolfram Research (2019), CrossingPolygon, Wolfram Language function, https://reference.wolfram.com/language/ref/CrossingPolygon.html.

CMS

Wolfram Language. 2019. "CrossingPolygon." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CrossingPolygon.html.

APA

Wolfram Language. (2019). CrossingPolygon. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CrossingPolygon.html

BibTeX

@misc{reference.wolfram_2025_crossingpolygon, author="Wolfram Research", title="{CrossingPolygon}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/CrossingPolygon.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_crossingpolygon, organization={Wolfram Research}, title={CrossingPolygon}, year={2019}, url={https://reference.wolfram.com/language/ref/CrossingPolygon.html}, note=[Accessed: 25-May-2025 ]}