Inmathematics, thewinding number orwinding index of aclosed curve in theplane around a givenpoint is aninteger representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve'snumber of turns. The winding number depends on theorientation of the curve, and it isnegative if the curve travels around the point clockwise.

Winding numbers are fundamental objects of study inalgebraic topology, and they play an important role invector calculus,complex analysis,geometric topology,differential geometry, andphysics (such as instring theory).

Suppose we are given a closed, oriented curve in thexy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin.

When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three.

Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be anyinteger. The following pictures show curves with winding numbers between −2 and 3:

${\displaystyle \cdots }$
	−2	−1	0
				${\displaystyle \cdots }$
	1	2	3

Let${\displaystyle \gamma :[0,1]\to \mathbb{C}\setminus \{a\}}$ be a continuous closed path on the complex plane minus one point. The winding number of${\displaystyle \gamma }$ around${\displaystyle a}$ is the integer

${\displaystyle \text{wind}(\gamma ,a)=s(1)-s(0),}$

where${\displaystyle (\rho ,s)}$ is the path written in polar coordinates, i.e. the lifted path through thecovering map

${\displaystyle p:{\mathbb{R}}_{>0}\times \mathbb{R}\to \mathbb{C}\setminus \{a\}:({\rho }_0,s_0)\mapsto a+{\rho }_0{e}^{i2\pi s_0}.}$

The winding number is well defined because of theexistence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of${\displaystyle p}$ are of the form${\displaystyle {\rho }_0\times (s_0+\mathbb{Z})}$ (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.

Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above:

A simplecombinatorial rule for defining the winding number was proposed byAugust Ferdinand Möbius in 1865^[1]and again independently byJames Waddell Alexander II in 1928.^[2]Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve).

Winding numbers play a very important role throughout complex analysis (cf. the statement of theresidue theorem). In the context ofcomplex analysis, the winding number of aclosed curve${\displaystyle \gamma }$ in thecomplex plane can be expressed in terms of the complex coordinatez =x +iy. Specifically, if we writez = re^iθ, then

${\displaystyle dz=e^{i\theta }dr+ir{e}^{i\theta }d\theta }$

and therefore

${\displaystyle \frac{dz}{z}=\frac{dr}{r}+id\theta =d[\ln r]+id\theta .}$

As${\displaystyle \gamma }$ is a closed curve, the total change in${\displaystyle \ln (r)}$ is zero, and thus the integral of$\frac{dz}{z}$ is equal to${\displaystyle i}$ multiplied by the total change in${\displaystyle \theta }$. Therefore, the winding number of closed path${\displaystyle \gamma }$ about the origin is given by the expression^[3]

${\displaystyle \frac{1}{2\pi i}{\oint }_{\gamma }\frac{dz}{z}.}$

More generally, if${\displaystyle \gamma }$ is a closed curve parameterized by${\displaystyle t\in [\alpha ,\beta ]}$, the winding number of${\displaystyle \gamma }$ about${\displaystyle z_0}$, also known as theindex of${\displaystyle z_0}$ with respect to${\displaystyle \gamma }$, is defined for complex${\displaystyle z_0\notin \gamma ([\alpha ,\beta ])}$ as^[4]

${\displaystyle {\mathrm{I}\mathrm{n}\mathrm{d}}_{\gamma }(z_0)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{d\zeta }{\zeta -z_0}=\frac{1}{2\pi i}{\int }_{\alpha }^{\beta }\frac{{\gamma }^{\prime }(t)}{\gamma (t)-z_0}dt.}$

This is a special case of the famousCauchy integral formula.

Some of the basic properties of the winding number in the complex plane are given by the following theorem:^[5]

Theorem.Let${\displaystyle \gamma :[\alpha ,\beta ]\to \mathbb{C}}$ be a closed path and let${\displaystyle \mathrm{\Omega }}$ be the set complement of the image of${\displaystyle \gamma }$, that is,${\displaystyle \mathrm{\Omega }:=\mathbb{C}\setminus \gamma ([\alpha ,\beta ])}$. Then the index of${\displaystyle z}$ with respect to${\displaystyle \gamma }$,$${\displaystyle {\mathrm{I}\mathrm{n}\mathrm{d}}_{\gamma }:\mathrm{\Omega }\to \mathbb{C},z\mapsto \frac{1}{2\pi i}{\oint }_{\gamma }\frac{d\zeta }{\zeta -z},}$$is (i) integer-valued, i.e.,${\displaystyle {\mathrm{I}\mathrm{n}\mathrm{d}}_{\gamma }(z)\in \mathbb{Z}}$ for all${\displaystyle z\in \mathrm{\Omega }}$; (ii) constant over each component (i.e., maximal connected subset) of${\displaystyle \mathrm{\Omega }}$; and (iii) zero if${\displaystyle z}$ is in the unbounded component of${\displaystyle \mathrm{\Omega }}$.

As an immediate corollary, this theorem gives the winding number of a circular path${\displaystyle \gamma }$ about a point${\displaystyle z}$. As expected, the winding number counts the number of (counterclockwise) loops${\displaystyle \gamma }$ makes around${\displaystyle z}$:

Corollary.If${\displaystyle \gamma }$ is the path defined by${\displaystyle \gamma (t)=a+r{e}^{int},0\le t\le 2\pi ,n\in \mathbb{Z}}$, then

Intopology, the winding number is an alternate term for thedegree of a continuous mapping. Inphysics, winding numbers are frequently calledtopological quantum numbers. In both cases, the same concept applies.

The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane ishomotopy equivalent to thecircle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps${\displaystyle S^1\to S^1:s\mapsto s^n}$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set ofhomotopy classes of maps from a circle to atopological space form agroup, which is called the firsthomotopy group orfundamental group of that space. The fundamental group of the circle is the additive group of theintegers,Z; and the winding number of a complex curve is just its homotopy class.

Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimesPontryagin index.

One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated at the beginning of this article has a winding number of 3, because the small loopis counted.

This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangentialGauss map.

This is called theturning number,rotation number,^[6]rotation index^[7] orindex of the curve, and can be computed as thetotal curvature divided by 2π.

Inpolygons, theturning number is referred to as thepolygon density. For convex polygons, and more generallysimple polygons (not self-intersecting), the density is 1, by theJordan curve theorem. By contrast, for a regularstar polygon {p/q}, the density isq.

Turning number cannot be defined for space curves asdegree requires matching dimensions. However, forlocally convex, closedspace curves, one can definetangent turning sign as${\displaystyle (-1{)}^d}$, where${\displaystyle d}$ is the turning number of thestereographic projection of itstangent indicatrix. Its two values correspond to the twonon-degenerate homotopy classes oflocally convex curves.^[8][9]

The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: theIshimori equation etc. Solutions of the last equations are classified by the winding number ortopological charge (topological invariant and/ortopological quantum number).

A point's winding number with respect to a polygon can be used to solve thepoint in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not.

Generally, theray casting algorithm is a better alternative to the PIP problem as it does not require trigonometric functions, contrary to the winding number algorithm. Nevertheless, the winding number algorithm can be sped up so that it too, does not require calculations involving trigonometric functions.^[10] The sped-up version of the algorithm, also known as Sunday's algorithm, is recommended in cases where non-simple polygons should also be accounted for.

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