Editing Winding number (section)

==Intuitive description==
[[Image:Winding Number Animation Small.gif|right|thumb|300px|An object traveling along the red curve makes two counterclockwise turns around the person at the origin.]]

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 &minus;2 and 3:

{| align="center" border=0 cellpadding=0
|-valign="center"
|<math>\cdots</math>
|align="center"|&nbsp;&nbsp;[[Image:Winding Number -2.svg|80px]]&nbsp;&nbsp;
|align="center"|&nbsp;&nbsp;[[Image:Winding Number -1.svg|80px]]&nbsp;&nbsp;
|align="center"|&nbsp;&nbsp;[[Image:Winding Number 0.svg|80px]]&nbsp;&nbsp;
|
|-valign="top" style="height:3em"
|
|align="center"|&minus;2
|align="center"|&minus;1
|align="center"|0
|
|-valign="center"
|
|align="center"|&nbsp;&nbsp;[[Image:Winding Number 1.svg|80px]]&nbsp;&nbsp;
|align="center"|&nbsp;&nbsp;[[Image:Winding Number 2.svg|80px]]&nbsp;&nbsp;
|align="center"|&nbsp;&nbsp;[[Image:Winding Number 3.svg|80px]]&nbsp;&nbsp;
|<math>\cdots</math>
|-valign="top"
|
|align="center"|1
|align="center"|2
|align="center"|3
|
|}