Simple Polygon - Weakly Simple Polygon

If a closed polygonal chain embedded in the plane divides it into two regions one of which is topologically equivalent to a disk, then the chain is called a weakly simple polygon. Informally, a weakly simple polygon is a polygon in which some sides can "touch" but cannot "cross over".

In the image on the left, ABCDEFGHJKLM is a weakly simple polygon with the color blue marking its interior.

In a more general definition of weakly simple polygons, they are the limits of sequences of simple polygons of the same combinatorial type, with the convergence under the Hausdorff metric. The "interior" can be empty. For example, referring to the image above, the polygonal chain ABCBA is a weakly simple polygon: it may be viewed as the limit of "squeezing" of the polygon ABCFGHA.

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.