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
Famous quotes containing the words formal and/or definition:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)