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.
Read more about this topic: Simple Polygon
Famous quotes containing the words weakly and/or simple:
“Let’s not quibble! I’m the foe of moderation, the champion of excess. If I may lift a line from a die-hard whose identity is lost in the shuffle, “I’d rather be strongly wrong than weakly right.””
—Tallulah Bankhead (1903–1968)
“I have often thought that if photography were difficult in the true sense of the term—meaning that the creation of a simple photograph would entail as much time and effort as the production of a good watercolor or etching—there would be a vast improvement in total output. The sheer ease with which we can produce a superficial image often leads to creative disaster.”
—Ansel Adams (1902–1984)