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We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions. We want to calculate the winding number of this curve around the origin. The most natural way to do it is to calculate the path integral:

$$\int_C \frac{-y\,dx+x\,dy}{x^2+y^2}$$

However, this integral turns out to be too complicated to calculate. What should we do now? Are there any efficient and strong methods to quickly and calculate the winding number?


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Without knowing the curve, it is very hard to tell what to do in your case... – Mariano Suárez-Alvarez Mar 21 '10 at 23:32
You mean, numerically? It ought to be a question of keeping track of how $(x,y)$ moves from one quadrant to the next – or previous, as the case may be. If you want a rigourous result, you need estimates on derivatives to make sure the curve can't do an undiscovered trip around the origin between time steps. – Harald Hanche-Olsen Mar 21 '10 at 23:33
@Harald Hanche-Olsen: No it needs to be done analytically. If you describe a general strategy, it would suffice. (A reference would be great.) @Mariano Suárez-Alvarez: What about if x(t) and y(t) are polynomials? – George G Mar 21 '10 at 23:39
That the computation is done numerically does not necessarily mean that the result is an aproximation; since you know that the result of the integral is an integer, aproximating it with an error less than, say, $1/4$ is enough to know the exact result... – Mariano Suárez-Alvarez Mar 21 '10 at 23:45
Your question is essentially open-ended, and you are always going to have a problem with a sufficiently evil curve... Unless you make precise what you want, so as to be able to see what an answer would be, this is not a great question, really! – Mariano Suárez-Alvarez Mar 22 '10 at 0:08
up vote 7 down vote accepted

This is simple if you can draw a picture of your curve. Find a direction so that your tangent is always moving as you pass through it. Count the number of tangents pointing in that direction with a sign. +1 if you are moving through the direction counterclockwise, and -1 if you are moving through the direction clockwise. The sum of the +1's and -1's is your winding number.

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The following well-known fact may be useful. If you continuously deform C into another loop C' without crossing the origin, then C' has the same winding number. (And the converse is true.)

In particular, you can reduce your problem to calculating the degree of a mapping S1 → S1.

Certainly, it's far from a general recipe.

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