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.



How To Calculate A Winding Number?
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.


