Is it against the definition of polygon for edges or vertices to overlap or being the same point or segment

Question

I think question shows what I'm after but I will try to add some more details. So there are two cases:

1. I have a polygon ABCDEC'(A). C' is a different point to C, but on a plane has exactly the same coordinates as C.

2. I have a polygon ABCDEC(A). C is a point on a plane which is used in the same polygon twice.

I couldn't find a definition which would prove those two to be incorrect. So my assumption is that they're perfectly valid polygons. Is it correct?

Answer 1

Here is a picture from Wikipedia showing some polygons. Note that three of them self intersect at a point (more than once for the regular star). Now, this diagram is not rigid at all when it comes to naming polygons, but I would say most definitions would agree that they are all polygons.... it is entirely dependent on your definition though. It is assumed that if you construct a polygon as you do in your post that you allow for this to occur. It is more a case by case basis... in some cases it is useful to break up an intersecting polygon into multiple non-intersecting polygons, and in some cases it is better to leave the polygon as one unit.

Answer 2

Try drawing out the points and the shapes themselves. If you draw out polygon 2 (i.e. draw points A-E and connect them as indicated), you'll see that this actually gives two shapes. So it's not necessarily against the definition of a polygon, but it does NOT produce a single polygon.