Talk:Simple polygon/Archive 1

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Archive 1

Are noncrossing intersections at vertices allowed simple polygons?

Let me try to represent this with text.

Would ABCDEC be a "simple polygon?" None of the edges intersect (except at the vertices, where they have to intersect). Should there be an additional restriction:

"Each vertex connects exactly 2 edges." With this restriction, ABCDEC would not be a simple polygon, since C connects 4 edges.

EDIT:

User sumthinelse

"iter praemium est"

I would say ABCDEC self-intersects on vertex C, and therefore is not simple. It might be worth an example image of special cases like this. Tom Ruen 22:08, 5 December 2006 (UTC)

Looking further, I see the USE of the definition, for dividing space into an inside and outside doesn't have a problem with "touching" edges or vertices. I wouldn't call such polygons simple, like above, but if they are considered such, perhaps a term like degenerate polygon would better apply.

The definition of degeneracy comes out between ABCDEC versus ABCEDC. Both look identical geometrically, but the orientation reverses. A nondegenerate simple polygon is well-defined without defining the path. —The preceding unsigned comment was added by Tomruen (talk • contribs) 22:29, 5 December 2006 (UTC).

A more interesting case would be like an Annulus (mathematics) - one convex polygon inside of another, with a double-edge connecting them, with the most interior region actually outside.

A--------B
|xxxxxxxx|
|xE---Fxx|
|x|   |xx|
|x|   |xx|
|xH-I-Gxx|
|xxx|xxxx|
D---J----C

ABCJIGFEHIJD

Is this simple? Tom Ruen 22:22, 5 December 2006 (UTC)

By definition, such crossings are not allowed. -- Cheers, Steelpillow (Talk) 08:59, 13 September 2008 (UTC)

I want to add, that there are no crossings in the depicted polygon. Only intersections, but no edge ever crosses the boundary of the polygon. --89.53.17.243 (talk) 19:28, 3 October 2009 (UTC)

That's definitely not allowed in a simple polygon. Sometimes it's useful to describe polygons that can have articulation points such as these but no crossings, but they need to be called something else, I think usually "weakly simple polygon". —David Eppstein (talk) 19:31, 3 October 2009 (UTC)

Firstly, if the figure is to be interpreted as a single polygon, the labelling is unhelpful. It does not make clear whether there are two distinct IJ edges, or whether ABCJIGFEHIJD is "joined at the hip", as it were, such that the abstract element IJ occurs twice in the same figure so that the figure is not actually a polygon at all. Better would be:

            A----------B
            |xxxxxxxxxx|
            |xH-----Gxx|
            |x|     |xx|
            |x|     |xx|
            |xJ-K,E-Fxx|
            |xxxx|xxxxx|
            M---L,D----C

ABCDEFGHJKLM

Now it is evident that there are two pairs of superimposed vertices - D,L and E,K - and one pair of superimposed edges - DE,KL - and we can at least be sure that we are dealing with a valid polygonal structure. Having said that, whether even this figure is a polygon or not depends on your definition of a "polygon". Some definitions exclude superimposed elements, others allow them. All would at least agree that a figure with superimposed elements is not simple. It is clearly a borderline case between simple and self-intersecting. Grünbaum coined the terms "coptic" for self-intersecting and "acoptic" for non-self-intersecting. It might be useful to say that this polygon is acoptic but not simple, but I'd have to check. -- Cheers, Steelpillow (Talk) 15:12, 4 October 2009 (UTC)

[1] - about polyhedra. Original text- Altenmann >t 17:48, 5 October 2009 (UTC)

See first link to the PDF: http://scholar.google.com/scholar?hl=es&q=Gr%C3%BCnbaum+acoptic&btnG=Buscar&lr=&as_ylo= But Grünbaum defined polyhedra that consist of simple polygons whose edges may overlap as acoptic, not polygons. So the equivalent definition for polygons would be... perhaps "acoptic polygons". The only problem is that noone coined that word yet. There are no acoptic polygons as much as there are no acoptic Sets --134.100.32.213 (talk) 16:30, 21 October 2009 (UTC)

Grünbaum uses the adjective "acoptic" in many papers. He defines it sufficiently generally for it to apply to polygons, polyhedra and polytopes in general. HTH 83.104.46.71 (talk) 12:05, 4 April 2010 (UTC)

ASCII-Art

Can someone please replace the ASCII art with a real Image? It looks very brocken depending on the fonts used... —Preceding unsigned comment added by Nefthy (talk • contribs) 10:35, 11 August 2010 (UTC)

: Is this better? : http://commons.wikimedia.org/wiki/File:SimplePolygon.png Quantumkayos (talk) 16:49, 27 August 2010 (UTC)

Could the lead be better?

Here is an alternative lead, based on belnapj's lead for the polygon article, which I think is more appropriate here because his definition equates to a simple polygon. I have edited it to try and keep the definition of a simple polygon in focus but also to keep the wider mathematical background within sight.

In geometry a simple polygon /ˈpɒlɪɡɒn/ is defined as a flat shape, consisting of straight, non-intersecting, line segments that are joined pair-wise to form a closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition being understood to define a polygon in general.

Mathematicians typically use "polygon" to refer only to the shape made up by the line segments, not the enclosed region, however some may use "polygon" to refer to a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain).^[1].

The definition given above ensures the following properties:

A polygon encloses a region (called its interior) and so it always has a measurable area.
The line segments that make-up a polygon (called sides or edges) meet only at their endpoints, called vertices (singular: vertex) or less formally "corners".
Exactly two edges meet at every vertex.
The number of edges always equals the number of vertices.
Two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge.

Simple polygons are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it. A simple polygon in the plane is topologically equivalent to a circle and its interior is topologically equivalent to a disk.

Any objections if I update the article accordingly? — Cheers, Steelpillow (Talk) 11:50, 26 November 2011 (UTC)

References

^ Grünbaum, B.; Convex polytopes 2nd Ed, Springer, 2003

Sorry for the delay in the response. Busy end-of-semester.

Looks good. I think you have done well with this. Thank you for your work.

By the way, You will find this interesting. Regarding whether a polygon includes its interior or not, you mentioned that the definition of a polyhedron is that the faces are polygons. This appears to be a laziness that has crept into some authors, because textual sources I have found use very carefully worded definitions that are consistent with simple polygons. One such example defines a polyhedron as: "A three-dimensional figure formed by flat surfaces that are bounded by polygons and joined in pairs along their sides. The polyhedron completely enclose a region of space, called its interior." 141.233.101.184 (talk) 17:06, 8 December 2011 (UTC)

Ooops, wasn't logged in. That last comment was mine. :) Dr. Belnap 17:07, 8 December 2011 (UTC) — Preceding unsigned comment added by Belnapj (talk • contribs)

OK, done. Defining a polyhedron is probably even more contentious than defining a polygon - as well as being careful to define the "polygons" it is constructed from! — Cheers, Steelpillow (Talk) 18:33, 8 December 2011 (UTC)

Dr. Belnap, the definition you quote for a polyhedron seems a little weird to me. It allows two cubes that touch at a single corner to count as a polyhedron (because its square faces meet in pairs and enclose a volume) but not two cubes that touch along an edge (because in this case there are four faces that do not meet in pairs). I think it's an example of an author trying to be careful and not being careful enough, rather than a good model to follow. As for this article: I think it could benefit from an example of something that is not a simple polygon, such as an antiparallelogram. —David Eppstein (talk) 19:24, 8 December 2011 (UTC)

Simple with Topology

Un polígono es simple si se puede establecer un homeomorfismo ( transformación topológica) entre él y una circunferencia. De modo que un punto interior en la región de un poligonal tiene su imagen en el interior del círculo correspondiente a la circunferencia. Se respeta el orden de los vértices que el mismo de las imágenes sobre la circunfrencia. Un punto exterior de la región poligonal tiene su imagen en el exterior del círculo.Hace rato que debiera enfocarse estos temas en la educación básica.Os pido disculpos, os invito a reflexionar.--X2y3 (talk) 03:30, 11 December 2015 (UTC)

Did you read our article? - üser:Altenmann >t 04:38, 12 December 2015 (UTC)

[1] Grünbaum, B.; Convex polytopes 2nd Ed, Springer, 2003

[1]