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Winding Numbers of Plane Curves

For any point Q not on a curve X , we may determine the number of times the curve

    WQ(t)=(X(t) -Q)/|X(t) -Q|
winds around the point Q . Clearly, this curve WQ(t) is a section of a circle centered at Q. The winding number is then defined as the total angular change in WQ(t) divided by 2{pi} . This number is also the number of times the curve X(t) winds around Q .

Winding Number Demo

In this demo, you may input a curve, in the control panel and then choose a point Q.

The demo displays the angular variation of the curve with respect to the point Q . The various lines with integer values n represent angles 2n{pi} . Thus, the function's final destination on the right-hand side of the window corresponds the winding number of the curve around Q .

    Investigate how the curve WQ(t) changes for various positions of Q . Try curves from the cardioid family.


Finis.