Talk:Simple polygon

I enjoy math, but I am a novice with it. I was wondering today if there is a proof that a simple polygon can always be made by connecting any set of dots on a plane, so I went online to try to find out. Is it possible that this sort of info could be provided on this page?

Thank you.

2Plus2Is4 (talk) 01:19, 27 June 2013 (UTC)

:Yes. For instance the solution to the traveling salesman problem is always a simple polygon. It should be possible to find this in a published source and add to the article. —David Eppstein (talk) 02:01, 27 June 2013 (UTC)

:He-he, no. If all points are collinear . Otherwise yes, and the first part of Graham scan will be a way faster way. Of related curiuosity, whether nlogn is the lower bound for complexity. - üser:Altenmann >t 04:29, 12 December 2015 (UTC) - üser:Altenmann >t 04:29, 12 December 2015 (UTC)

::In what model of computation? If the coordinates are integers and integer sorting is allowed, then no, you can do it faster. (Split the points into the subset above and below the line segment between leftmost and rightmost, and then connect each subset in sorted order.) However, in models of computation where convex hulls take $\backslash Omega(n\backslash log\; n)$ time (e.g. algebraic computation trees) constructing a simple polygon takes the same time, because it is known how to go from a simple polygon to the convex hull in linear time. —David Eppstein (talk) 04:40, 12 December 2015 (UTC)

Our section currently states "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." But this cannot be correct: if the chain is embedded, it cannot touch itself (that's what the definition of "embedding" is), so it is simple (not just weakly simple) and by the Jordan curve theorem always divides the plane into two parts. And the associated illustration, which is supposed to match this "definition", does not show an embedded chain, but one that has self-intersections. So what is the correct form of the definition that is intended here? —David Eppstein (talk) 04:55, 12 December 2015 (UTC)

:I changed this part to more accurately reflect what its source actually says. However, I suspect there are other sources out there with definitions that differ from the two given here in subtle or less-subtle ways, some of which may resemble the part I replaced. If so they should be added. —David Eppstein (talk) 05:45, 12 December 2015 (UTC)

It may be worth adding a discussion at the end of polygons on the sphere. These are practically important for GIS etc. have some interesting differences from planar polygons; for example, there is no a priori reason to declare one side or the other as the "interior". In particular things get tricky when a polygon does not fit within a hemisphere, e.g. such polygons don't have an easily defined convex hull. –jacobolus (t) 03:25, 18 July 2023 (UTC)

At some point a while back I looked up a whole bunch of analysis textbooks to compare their definitions/usage of the terms "domain" and "region", as evidence for figuring out what the top couple sections of domain should say. (The "historical notes" section there is not really right; Caratheodory definitely did not introduce this concept.) From everything I can tell, these terms are largely used interchangeably across mathematical subdisciplines to mean "connected open set in a topological space", which is a precise formalization of a casual term meaning something like "the inside or outside of a shape". (Some sources, rather confusingly in my opinion, specifically define the two to mean subtly different things.)

The reason I think it's helpful to wikilink terms like this, even if the link might sometimes fly over the head of some of the intended audience, is that many readers don't actually know that this is what "region" informally means in mathematics. The article circle (whose text I didn't write), says:

: Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.

Just yesterday someone tried to change it to say that the circle divides the plane into 3 regions: the interior, the circle itself, and the exterior. To someone who is not familiar with the customary techical meaning of the jargon word "region", it's not inherently obvious that a circle (or a simple polygon) can't be a "region". Obviously the word "region" originally comes by analogy with the geographical / cartographical term region, and you'd think people would roughly have a sense of the distinction between a region – say an administrative district – and its border or boundary, but when a word is transplanted from one context to another like that, it's helpful to be able to point to a definition to avoid confusion and give readers something to look into if they are curious to learn further details.

The better solution in my opinion would be to add some material at the top of domain which is more accessible to non-technical readers, along with a picture or two and some examples. –jacobolus (t) 08:01, 9 September 2023 (UTC)

For a simple polygon, the turning number or "density" is always equal to 1. Perhaps this should be worked into the paragraph about being a Jordan curve, or into the paragraph about internal and external angles. –jacobolus (t) 17:21, 17 September 2023 (UTC)

:This is the same thing as the sum of external angles, but for smooth curves. Polygons are not smooth curves, so we should not link to articles defined to require smoothness. —David Eppstein (talk) 19:21, 17 September 2023 (UTC)

::Eh, the concept (like total curvature) is most naturally defined first for polygons then generalized to arbitrary closed curves (which can be taken as a smooth limit of polygons as the vertices get infinitely close together / sides get infinitesimally short – as familiar in every other concept related to smooth shapes and motions built on calculus, this limit turns a sum into an integral; if we allow curvature as a function of arclength to be a distribution we can easily account for closed curves with cusps, etc.). I think the version in Wikipedia overconstrains and does a poor job explaining the concept, but perhaps that's inherited from available sources just doing a bad job.

::Despite leading with a limiting definition, turning number does mention polygons. Then density (polytope)#Polygons is entirely about polygons. –jacobolus (t) 19:45, 17 September 2023 (UTC)

:::We could say something like "The fact that all simple polygons have the same sum of external angles can be generalized to simple closed curves, which all have the same turning number.", at the end of the paragraph on external angles. But we would need a published source making that specific connection. —David Eppstein (talk) 20:46, 17 September 2023 (UTC)

::::This paper seems to have some good historical survey: https://www.ams.org/journals/tran/1990-322-01/S0002-9947-1990-1024774-2/S0002-9947-1990-1024774-2.pdf – as I would have expected the polygon concept seems to have come first. –jacobolus (t) 21:53, 17 September 2023 (UTC)

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