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July 18

Polygon triangulation

The article on Polygon triangulation states that "In the strict sense, these triangles may have vertices only at the vertices of P." I want to know whether it is always possible to triangulate a polygon in this fashion, and if so, why. Thanks---Shahab (talk) 12:54, 18 July 2012 (UTC)[reply]

Provided, for any polygon, it is always possible to draw at least one straight line between two vertices that lies entirely inside the polygon, then any polygon can be dissected into two smaller ones, and the process continued until only triangles are left. So, for the triangulation not to be possible, there would have to be a polygon for which it wasn't possible to draw such a straight line. 86.160.212.146 (talk) 20:55, 18 July 2012 (UTC)[reply]

Okay. Basically if the polygon is convex. But I don't know of a necessary condition.-Shahab (talk) 15:12, 19 July 2012 (UTC)[reply]

I don't exactly understand what you mean by that, but this argument works equally for convex and concave polygons. I cannot conceive of any polygon, concave or convex, where it is not possible to draw at least one line connecting vertices that lies entirely within the polygon. I think it is certain that such a polygon cannot exist,* but I don't know how to actually prove it. 86.146.110.153 (talk) 19:29, 19 July 2012 (UTC) * Other than a triangle, obviously...[reply]

Here's a (messy) proof for if the polygon is not convex: The polygon has at least one "inner" corner, call it B, and suppose the adjacent vertices are A and C. Sweep a ray from B along all the angles between BA and BC through the interior of the polygon. For any given angle, the ray will hit the boundary of the polygon somewhere, and just consider the first such point it hits. If any of the rays hit a vertex, you're done. If none of them hit a vertex, then they all must hit the same face (this part would take some effort to flesh out). This is impossible because the interior angle at B is greater than 180°. I hope there's a nicer proof though. Rckrone (talk) 06:23, 20 July 2012 (UTC)[reply]

It is always possible to find a triangle which is entirely contained in a given polygon and has all its vertices in the polygon vertices. Start with any convex angle of the polygon, like

\angle ABC

in this polygon. Consider a

\triangle ABC

:

if a line segment $AC$ lies entirely in the polygon, cut the triangle off the polygon,
otherwise there are some polygon vertices in the triangle – find the one of them (e.g. $G$ ) such that the line segment $BG$ lies entirely in the polygon, and recursively consider $\triangle GBC$ .

In a finite sequence of such steps you find a desired triangle.

Removing it may, however, split the polygon into pieces touching only in their common vertex (or vertices). Then proceed with each piece separately. --CiaPan (talk) 07:02, 20 July 2012 (UTC)[reply]

Collective term for interpolation and extrapolation

Interpolation and extrapolation are obviously closely linked, so what is the collective term for them? Obviously you could describe them with something very generic like "prediction", but is there a more specific word? It seems odd that e.g. linear interpolation and linear extrapolation have exactly the same formula (if y > x and 0 < p < 1 then we can interpolate a fraction p between x and y using lerp (x, y, p) = x + p(y-x) = (1-p)x + py ... but if p is greater than 1 or below zero this is actually just linear extrapolation) but for them not to have a common name. The obvious answer is "polation" of course, but the dictionaries show no such word (though I've come across some computer scientists using it). For general results about both processes, I've seen academic papers with "inter/extrapolation" in the title. It's really crying out for a single word that describes both! But is there one? ManyQuestionsFewAnswers (talk) 21:15, 18 July 2012 (UTC)[reply]

I don't think there is such a word, and I don't think "prediction" is a very good choice, at least in the usual sense of guessing what will happen in the future, as that's really only an extrapolation forward in time. However, there's often a lack of a single word for paired concepts. Is there a word that means either "enter" or "exit" ? StuRat (talk) 21:20, 18 July 2012 (UTC)[reply]

"Prediction" would be a poor choice in my opinion, but I wanted to get it in there so that nobody else said it! It's true that quite a lot of paired concepts lack a single word. But I can't think of another case where two mathematical concepts share exactly the same formula without having a common word - mathematicians usually like to generalize too much for that! ManyQuestionsFewAnswers (talk) 21:45, 18 July 2012 (UTC)[reply]

Alternate definition?

Would the standard definition of Regular Polyhedra (including the Regular Star Polyhedra) be equivalent to the following: 1) All edges are the same length and 2) All faces have the same area. If not, can someone please give me a counter example?

Do you mean like Rhombohedron? 86.160.212.146 (talk) 20:58, 18 July 2012 (UTC)[reply]

Thanx. The Rhombic dodecahedron also works as a counter example.Naraht (talk) 13:45, 19 July 2012 (UTC)[reply]