How To Calculate A Winding Number? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:15:12Z http://mathoverflow.net/feeds/question/18960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18960/how-to-calculate-a-winding-number How To Calculate A Winding Number? George G 2010-03-21T23:18:54Z 2010-03-22T22:32:09Z <p>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: </p> <p><code>$$\int_C \frac{-y\,dx+x\,dy}{x^2+y^2}$$</code></p> <p>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?</p> <p>Thanks.</p> http://mathoverflow.net/questions/18960/how-to-calculate-a-winding-number/18965#18965 Answer by AlB for How To Calculate A Winding Number? AlB 2010-03-21T23:59:02Z 2010-03-21T23:59:02Z <p>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.)</p> <p>In particular, you can reduce your problem to calculating the degree of a mapping S<sup>1</sup> → S<sup>1</sup>.</p> <p>Certainly, it's far from a general recipe.</p> http://mathoverflow.net/questions/18960/how-to-calculate-a-winding-number/18968#18968 Answer by Charlie Frohman for How To Calculate A Winding Number? Charlie Frohman 2010-03-22T00:31:14Z 2010-03-22T00:31:14Z <p>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.</p>