# How To Calculate A Winding Number?

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?

Thanks.

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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