What Does "Inside" Mean?

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Consider a closed directed curve (or loop) in the plane. If this loop is simple, that is, it does not intersect itself, then however complicated it might be, it will divide the plane into two sets: an inside and an outside. In the case of a non-simple loop, like the curve shown, it is no longer obvious which points are to be considered inside the loop and which outside. The winding number concept clarifies that distinction and plays a crucial role in understanding multifunctions and complex integration.


The winding number of a loop around a point (represented by a draggable red disk) is the net number of revolutions of a point around as it traverses . The "inside" of can be defined as those points for which the winding number is an odd number (depicted in cyan).

This Demonstration lets you drag any of the yellow disks, which act as control points to change the shape of . Also, there is an optional movable arrow (moving its tip) going from outward that shows the changes in unit increments or decrements of the winding number as we move away from . So, to find out if a point is inside a loop, consider an outward line starting from it; if the number of intersections is odd, the point is inside.


Contributed by: Jaime Rangel-Mondragon (September 2018)
Open content licensed under CC BY-NC-SA



[1] T. Needham, Visual Complex Analysis, New York: Oxford University Press, 1997.


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