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Non-simple weakly simple polygons arise in computer graphics and CAD as a computer representation of polygonal regions with holes: for each hole a " cut " is created to connect it to an external boundary.
Referring to the image above, ABCM is an external boundary of a planar region with a hole FGHJ.
The cut ED connects the hole with the exterior and is traversed twice in the resulting weakly simple polygonal representation.

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