Winding number
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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 (such as in string theory).
Intuitive description
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 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 any integer. The following pictures show curves with winding numbers between −2 and 3:
−2 | −1 | 0 | ||
1 | 2 | 3 |
Formal definition
Let be a continuous closed path on the plane minus one point. The winding number of around is the integer
where is the path written in polar coordinates, i.e. the lifted path through the covering map
The winding number is well defined because of the existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of are of the form (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.
Alternative definitions
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:
Alexander numbering
A simple
Differential geometry
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:
Which is found by differentiating the following definition for θ:
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:
The
Complex analysis
Winding numbers play a very important role throughout complex analysis (c.f. the statement of the
and therefore
As is a closed curve, the total change in is zero, and thus the integral of is equal to multiplied by the total change in . Therefore, the winding number of closed path about the origin is given by the expression[3]
More generally, if is a closed curve parameterized by , the winding number of about , also known as the index of with respect to , is defined for complex as[4]
This is a special case of the famous
Some of the basic properties of the winding number in the complex plane are given by the following theorem:[5]
Theorem. Let be a closed path and let be the set complement of the image of , that is, . Then the index of with respect to ,
As an immediate corollary, this theorem gives the winding number of a circular path about a point . As expected, the winding number counts the number of (counterclockwise) loops makes around :
Corollary. If is the path defined by , then
Topology
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 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 is
Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes
Turning number
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 loop is counted.
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.
This is called the turning number, rotation number,[6] rotation index[7] or index of the curve, and can be computed as the total curvature divided by 2π.
Polygons
In
Space curves
Turning number cannot be defined for space curves as
Winding number and Heisenberg ferromagnet equations
The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the
Applications
Point in polygon
A point's winding number with respect to a polygon can be used to solve the point in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not.
Generally, the ray 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.
See also
- Argument principle
- Coin rotation paradox
- Linking coefficient
- Nonzero-rule
- Polygon density
- Residue theorem
- Schläfli symbol
- Topological degree theory
- Topological quantum number
- Twist (mathematics)
- Wilson loop
- Writhe
References
- ^ Möbius, August (1865). "Über die Bestimmung des Inhaltes eines Polyëders". Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Klasse. 17: 31–68.
- JSTOR 1989123.
- ^ Weisstein, Eric W. "Contour Winding Number". MathWorld. Retrieved 7 July 2022.
- ISBN 0-07-054235-X.
- ISBN 0-07-054234-1.
- ^ Abelson, Harold (1981). Turtle Geometry: The Computer as a Medium for Exploring Mathematics. MIT Press. p. 24.
- ISBN 0-13-212589-7.
- S2CID 116999463.
- S2CID 252274622.
- ^ Sunday, Dan (2001). "Inclusion of a Point in a Polygon". Archived from the original on 26 January 2013.