WindingCountCopy to clipboard.
✖
WindingCount
gives the count of the number of times a closed curve winds around a point p.
Details and Options
- WindingCount is also known as winding number.
- WindingCount is typically used to define a polygon from self-intersecting closed curves.
- A counterclockwise revolution has a value 1 and a clockwise revolution has a value -1. For several contours, the number of revolutions around p is the sum of the number of revolutions for each contour.
- 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.
- For a polygon poly, WindingCount[poly,p] gives the number of times the polygon boundary curves wind around the point p.
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Count how many times a closed curve winds around a point:
https://wolfram.com/xid/0h2ri3fb9e-dpae4v
https://wolfram.com/xid/0h2ri3fb9e-2b0d1m
https://wolfram.com/xid/0h2ri3fb9e-th00ia
Scope (2)Survey of the scope of standard use cases
Count how many times a closed curve winds around a point:
https://wolfram.com/xid/0h2ri3fb9e-6fpmk2
https://wolfram.com/xid/0h2ri3fb9e-zsuhb5
https://wolfram.com/xid/0h2ri3fb9e-e9q408
Count how many times a polygon boundary curve winds around a point:
https://wolfram.com/xid/0h2ri3fb9e-pjd2d1
https://wolfram.com/xid/0h2ri3fb9e-4hqpq3
https://wolfram.com/xid/0h2ri3fb9e-ckaagt
Applications (2)Sample problems that can be solved with this function
Find the polygon density of regular star polygons:
https://wolfram.com/xid/0h2ri3fb9e-o1937e
https://wolfram.com/xid/0h2ri3fb9e-7yte82
https://wolfram.com/xid/0h2ri3fb9e-qbgnx6
https://wolfram.com/xid/0h2ri3fb9e-oe71wo
Color points by the winding count of the point around the given contour:
https://wolfram.com/xid/0h2ri3fb9e-tkpxgr
https://wolfram.com/xid/0h2ri3fb9e-11p3t1
https://wolfram.com/xid/0h2ri3fb9e-rv8poi
https://wolfram.com/xid/0h2ri3fb9e-dm6jz1
Properties & Relations (1)Properties of the function, and connections to other functions
CrossingCount is an alternate count function:
https://wolfram.com/xid/0h2ri3fb9e-0h89de
https://wolfram.com/xid/0h2ri3fb9e-14smh2
Possible Issues (1)Common pitfalls and unexpected behavior
A polygon boundary curve is always given in the counterclockwise orientation with no overlapping:
https://wolfram.com/xid/0h2ri3fb9e-sjsx30
https://wolfram.com/xid/0h2ri3fb9e-rba327
https://wolfram.com/xid/0h2ri3fb9e-inmg47
https://wolfram.com/xid/0h2ri3fb9e-474tjc
Wolfram Research (2019), WindingCount, Wolfram Language function, https://reference.wolfram.com/language/ref/WindingCount.html.
Text
Wolfram Research (2019), WindingCount, Wolfram Language function, https://reference.wolfram.com/language/ref/WindingCount.html.
Wolfram Research (2019), WindingCount, Wolfram Language function, https://reference.wolfram.com/language/ref/WindingCount.html.
CMS
Wolfram Language. 2019. "WindingCount." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WindingCount.html.
Wolfram Language. 2019. "WindingCount." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WindingCount.html.
APA
Wolfram Language. (2019). WindingCount. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WindingCount.html
Wolfram Language. (2019). WindingCount. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WindingCount.html
BibTeX
@misc{reference.wolfram_2024_windingcount, author="Wolfram Research", title="{WindingCount}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/WindingCount.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_windingcount, organization={Wolfram Research}, title={WindingCount}, year={2019}, url={https://reference.wolfram.com/language/ref/WindingCount.html}, note=[Accessed: 08-January-2025
]}