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CrossingCount
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CrossingCount

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CrossingCount[contour,p]

gives a count of the number of times a ray starting from the point p crosses the closed curve contour.

Details and Options

  • CrossingCount is also known as crossing number.
  • CrossingCount is typically used to determine the inclusion of a point in a polygon.
  • If the closed curve contour is self‐intersecting, CrossingCount[contour,p] gives the minimal count among all rays starting from the point p.
  • The closed curve contour is typically a collection of line segments Line[{{p1,p2},…,{pn-1,pn},{pn,p1}}] and must lie in a plane.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Count how many times a ray starting from the point crosses the closed-curve contour:

In[1]:=1
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https://wolfram.com/xid/0dqupcr75oy-cacgxz
In[2]:=2
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https://wolfram.com/xid/0dqupcr75oy-e3zlpg
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqupcr75oy-e9q408
Out[3]=3

Scope  (3)Survey of the scope of standard use cases

Count how many times a ray starting from the point crosses the closed-curve contour:

In[1]:=1
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https://wolfram.com/xid/0dqupcr75oy-6fpmk2
In[2]:=2
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https://wolfram.com/xid/0dqupcr75oy-zsuhb5
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqupcr75oy-3dno4m
Out[3]=3

Polygon boundary curves:

In[1]:=1
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https://wolfram.com/xid/0dqupcr75oy-ell501
In[2]:=2
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https://wolfram.com/xid/0dqupcr75oy-4syxlc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqupcr75oy-4stb6a
Out[3]=3

Polygons with GeoGridPosition:

In[1]:=1
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https://wolfram.com/xid/0dqupcr75oy-ipsiqu
In[2]:=2
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https://wolfram.com/xid/0dqupcr75oy-yjciq7
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqupcr75oy-cn3c0c
Out[3]=3

Applications  (1)Sample problems that can be solved with this function

Color points based on the crossing count of a point in the given polygon:

In[13]:=13
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https://wolfram.com/xid/0dqupcr75oy-51wvlh
In[14]:=14
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https://wolfram.com/xid/0dqupcr75oy-yvi4jj
In[15]:=15
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https://wolfram.com/xid/0dqupcr75oy-0v95f7
In[16]:=16
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https://wolfram.com/xid/0dqupcr75oy-kzpu27
Out[16]=16

Properties & Relations  (1)Properties of the function, and connections to other functions

WindingCount is an alternate count function:

In[1]:=1
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https://wolfram.com/xid/0dqupcr75oy-0h89de
In[2]:=2
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https://wolfram.com/xid/0dqupcr75oy-14smh2
Out[2]=2
Wolfram Research (2019), CrossingCount, Wolfram Language function, https://reference.wolfram.com/language/ref/CrossingCount.html.
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Wolfram Research (2019), CrossingCount, Wolfram Language function, https://reference.wolfram.com/language/ref/CrossingCount.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_crossingcount, author="Wolfram Research", title="{CrossingCount}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/CrossingCount.html}", note=[Accessed: 30-April-2025 ]}

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@misc{reference.wolfram_2025_crossingcount, author="Wolfram Research", title="{CrossingCount}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/CrossingCount.html}", note=[Accessed: 30-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_crossingcount, organization={Wolfram Research}, title={CrossingCount}, year={2019}, url={https://reference.wolfram.com/language/ref/CrossingCount.html}, note=[Accessed: 30-April-2025 ]}

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@online{reference.wolfram_2025_crossingcount, organization={Wolfram Research}, title={CrossingCount}, year={2019}, url={https://reference.wolfram.com/language/ref/CrossingCount.html}, note=[Accessed: 30-April-2025 ]}