Formal Definition
A curve in the xy plane can be defined by parametric equations:
If we think of the parameter t as time, then these equations specify the motion of an object in the plane between t = 0 and t = 1. The path of this motion is a curve as long as the functions x(t) and y(t) are continuous. This curve is closed as long as the position of the object is the same at t = 0 and t = 1.
We can define the winding number of such a curve using the polar coordinate system. Assuming the curve does not pass through the origin, we can rewrite the parametric equations in polar form:
The functions r(t) and θ(t) are required to be continuous, with r > 0. Because the initial and final positions are the same, θ(0) and θ(1) must differ by an integer multiple of 2π. This integer is the winding number:
This defines the winding number of a curve around the origin in the xy plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point p.
Read more about this topic: Winding Number
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