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| :''The term ''winding number'' may also refer to the [[rotation number]] of an [[iterated map]].''
| | I’m Isobel from Herzogenbuchsee doing my final year engineering in Nutritional Sciences. I did my schooling, secured 85% and hope to find someone with same interests in Skydiving.<br><br>My page [http://invillas.com/blog/for-dining-in-italy-luxury-villas-is-a-top-choice/ Window Installatio] |
| [[Image:Winding Number Around Point.svg|thumb|right|250px|This curve has winding number two around the point ''p''.]]
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| In [[mathematics]], the '''winding number''' of a closed [[curve]] in the [[plane (mathematics)|plane]] around a given [[point (mathematics)|point]] is an [[integer]] representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the [[curve orientation|orientation]] of the curve, and is [[negative number|negative]] if the curve travels around the point clockwise.
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| Winding numbers are fundamental objects of study in [[algebraic topology]], and they play an important role in [[vector calculus]], [[complex analysis]], [[geometric topology]], [[differential geometry]], and [[physics]], including [[string theory]].
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| ==Intuitive description==
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| [[Image:Winding Number Animation Small.gif|right|thumb|300px|An object traveling along the red curve makes two counterclockwise turns around the person at the origin.]]
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| Suppose we are given a closed, oriented curve in the ''xy'' 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 [[Turn (geometry)|turns]] that the object makes around the origin.
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| When counting the total number of [[Turn (geometry)|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.
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| 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 any [[integer]]. The following pictures show curves with winding numbers between −2 and 3:
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| {| align="center" border=0 cellpadding=0
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| |-valign="center"
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| |<math>\cdots</math>
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| |align="center"| [[Image:Winding Number -2.svg|80px]]
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| |align="center"| [[Image:Winding Number -1.svg|80px]]
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| |align="center"| [[Image:Winding Number 0.svg|80px]]
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| |-valign="top" style="height:3em"
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| |align="center"|−2
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| |align="center"|−1
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| |align="center"|0
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| |-valign="center"
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| |align="center"| [[Image:Winding Number 1.svg|80px]]
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| |align="center"| [[Image:Winding Number 2.svg|80px]]
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| |align="center"| [[Image:Winding Number 3.svg|80px]]
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| |<math>\cdots</math>
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| |-valign="top"
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| |align="center"|1
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| |align="center"|2
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| |align="center"|3
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| |}
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| ==Formal definition==
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| A curve in the ''xy'' plane can be defined by [[parametric equation]]s:
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| :<math>x = x(t)\quad\text{and}\quad y=y(t)\qquad\text{for }0 \leq t \leq 1.</math>
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| If we think of the parameter ''t'' as time, then these equations specify the motion of an object in the plane between {{nowrap|1= ''t'' = 0}} and {{nowrap|1= ''t'' = 1}}. The path of this motion is a curve as long as the [[Function (mathematics)|functions]] ''x''(''t'') and ''y''(''t'') are [[continuous function|continuous]]. This curve is closed as long as the position of the object is the same at {{nowrap|1= ''t'' = 0}} and {{nowrap|1= ''t'' = 1}}.
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| We can define the '''winding number''' of such a curve using the [[polar coordinate system]]. Assuming the curve does not pass through the origin, we can rewrite the parametric equations in polar form:
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| :<math>r = r(t)\quad\text{and}\quad \theta = \theta(t)\qquad\text{for }0 \leq t \leq 1.</math>
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| The functions ''r''(''t'') and ''θ''(''t'') are required to be continuous, with {{nowrap| ''r'' > 0}}. Because the initial and final positions are the same, ''θ''(0) and ''θ''(1) must differ by an integer multiple of 2''π''. This integer is the winding number:
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| :<math>\text{winding number} = \frac{\theta(1) - \theta(0)}{2\pi}.</math>
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| This defines the winding number of a curve around the origin in the ''xy'' plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point ''p''.
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| ==Alternative definitions==
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| 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:
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| ===Differential geometry===
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| In [[differential geometry]], parametric equations are usually assumed to be [[differentiable]] (or at least piecewise differentiable). In this case, the polar coordinate ''θ'' is related to the rectangular coordinates ''x'' and ''y'' by the equation:
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| :<math>d\theta = \frac{1}{r^2} \left( x\,dy - y\,dx \right)\quad\text{where }r^2 = x^2 + y^2.</math>
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| By the [[fundamental theorem of calculus]], the total change in ''θ'' is equal to the [[integral]] of ''dθ''. We can therefore express the winding number of a differentiable curve as a [[line integral]]:
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| :<math>\text{winding number} = \frac{1}{2\pi} \oint_C \,\frac{x}{r^2}\,dy - \frac{y}{r^2}
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| \,dx.</math>
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| The [[one-form]] ''dθ'' (defined on the complement of the origin) is [[closed and exact differential forms|closed]] but not exact, and it generates the first [[de Rham cohomology]] group of the [[punctured plane]]. In particular, if ''ω'' is any closed differentiable one-form defined on the complement of the origin, then the integral of ''ω'' along closed loops gives a multiple of the winding number.
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| ===Complex analysis===
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| In [[complex analysis]], the winding number of a closed curve ''C'' in the [[complex plane]] can be expressed in terms of the complex coordinate {{nowrap|1= ''z'' = ''x'' + ''iy''}}. Specifically, if we write ''z'' = ''re''<sup>''iθ''</sup>, then
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| :<math>dz = e^{i\theta} dr + ire^{i\theta} d\theta\!\,</math>
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| and therefore
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| :<math>\frac{dz}{z} \;=\; \frac{dr}{r} + i\,d\theta \;=\; d[ \ln r ] + i\,d\theta.</math>
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| The total change in ln(''r'') is zero, and thus the integral of ''dz'' ⁄ ''z'' is equal to ''i'' multiplied by the total change in ''θ''. Therefore:
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| :<math>\text{winding number} = \frac{1}{2\pi i} \oint_C \frac{dz}{z}.</math>
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| More generally, the winding number of ''C'' around any complex number ''a'' is given by
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| :<math>\frac{1}{2\pi i} \oint_C \frac{dz}{z - a}.</math>
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| This is a special case of the famous [[Cauchy integral formula]]. Winding numbers play a very important role throughout complex analysis (c.f. the statement of the [[residue theorem]]).
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| ===Topology===
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| In [[topology]], the winding number is an alternate term for the [[degree of a continuous mapping]]. In [[physics]], winding numbers are frequently called [[topological quantum number]]s. In both cases, the same concept applies.
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| The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is [[homotopy equivalent]] to the [[circle]], 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 <math>S^1 \to S^1 : s \mapsto s^n</math>, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of [[homotopy class]]es of maps from a circle to a [[topological space]] form a [[Group (mathematics)|group]], which is called the first [[homotopy group]] or [[fundamental group]] of that space. The fundamental group of the circle is the group of the [[integers]], '''Z'''; and the winding number of a complex curve is just its homotopy class.
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| Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes [[Pontryagin index]].
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| ===Polygons===
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| [[File:Star polygon 9 4.png|thumb|The boundary of the regular [[Enneagram (geometry)|Enneagram]] {9/4} winds around its centre 4 times, so it has a [[polygon density|density]] of 4.]]
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| In [[polygon]]s, the winding number is referred to as the [[polygon density]]. For convex polygons, and more generally [[simple polygon]]s (not self-intersecting), the density is 1, by the [[Jordan curve theorem]]. By contrast, for a regular [[star polygon]] {''p''/''q''}, the density is ''q''.
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| ==Turning number==
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| 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 on the right has a winding number of 4 (or −4), because the small loop ''is'' counted.
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| This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential [[Gauss map]].
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| This is called the '''turning number''', and can be computed as the [[total curvature]] divided by 2''π''.
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| ==Winding number and Heisenberg ferromagnet equations==
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| Finally, note that the winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the [[Ishimori equation]] etc. Solutions of the last equations are classified by the winding number or [[topological charge]] ([[topological invariant]] and/or [[topological quantum number]]).
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| ==See also==
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| * [[Argument principle]]
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| * [[Linking coefficient]]
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| * [[Polygon density]]
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| * [[Residue theorem]]
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| * [[Topological degree theory]]
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| * [[Topological quantum number]]
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| * [[Wilson loop]]
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| * [[Nonzero-rule]]
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| ==External links==
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| *{{planetmath reference|title=Winding number|id=3291}}
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| [[Category:Algebraic topology]]
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| [[Category:Complex analysis]]
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| [[Category:Differential geometry]]
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I’m Isobel from Herzogenbuchsee doing my final year engineering in Nutritional Sciences. I did my schooling, secured 85% and hope to find someone with same interests in Skydiving.
My page Window Installatio