# WindingCount

WindingCount[contour,p]

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

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## Basic Examples(1)

Count how many times a closed curve winds around a point:

## Scope(2)

Count how many times a closed curve winds around a point:

Count how many times a polygon boundary curve winds around a point:

## Applications(2)

Find the polygon density of regular star polygons:

Color points by the winding count of the point around the given contour:

## Properties & Relations(1)

CrossingCount is an alternate count function:

## Possible Issues(1)

A polygon boundary curve is always given in the counterclockwise orientation with no overlapping:

The winding count:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2023_windingcount, author="Wolfram Research", title="{WindingCount}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/WindingCount.html}", note=[Accessed: 28-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_windingcount, organization={Wolfram Research}, title={WindingCount}, year={2019}, url={https://reference.wolfram.com/language/ref/WindingCount.html}, note=[Accessed: 28-September-2023 ]}