Talk:Ideal point

Discussion that led to this page

This is the discussion from Talk:Point at infinity#Hyperbolic geometry section that led to this page. Mostly historical interest. WillemienH (talk) 22:02, 22 June 2015 (UTC)

The section Point at infinity § Hyperbolic geometry is premised on a confusion: between points points at infinity and omega points. I suspect that some authors may define the term point at infinity differently, but whatever the case might be, this article should define the term in one way and then use it to mean something incompatible, especially without redefining it. Since hyperbolic geometry is studied in projective geometry, and in particular is a projective geometry (in the same sense that affine, Euclidian, elliptic and many others are), conflicting terminology must be avoided. IMO, the omega points of hyperbolic geometry are not points at infinity as defined in the section on Projective geometry (the term really only makes sense in the affine geometries). I think that we should examine the definitions of "point at infinity" that occur and resolve this; in particular, we need to be clear on the terms point of infinity, ideal point, and omega point. —Quondum 13:30, 16 June 2015 (UTC)

I was thinking of splitting the section on hyperbolic geometry out to a new page ideal point (which is now an redrect page to this page )

I don't think the points have the same meaning in hyperbolic geometry as in the other geometries (in the other geometries lines have one point at infinity, while in hyperbolic geometry they have two) Not sure about the difference between omega point and ideal point. Can you add the banners to this idea (if you support it)

Also i found the lead confusing is it about the one dimensional case or about the 2 dimensional case? WillemienH (talk) 20:39, 16 June 2015 (UTC)

I still need to look at references to say anything definitive about the use of the terms point at infinity, omega point and ideal point. Once I can figure out widely used definitions for the terms, I should be able to look into this. I'd wait on the splitting.

The lead is confusing. It is speaking of the one-dimensional case: the Riemann sphere is a one-dimensional space over the complex numbers, specifically the projective completion of the complex line (which gets called a plane because it is parametrized by the complex numbers, but in technically it is a complex line ;P ). —Quondum 21:42, 16 June 2015 (UTC)

Browsing about, it seems to me that point at infinity and ideal point are quite extensively used in both the affine and hyperbolic families of geometries. I do not see omega point, though. The ideal points in an affine geometry form a flat (line, plane etc.) and there is one on each line, whereas in hyperbolic geometry the ideal points form a conic, and there are two on each line. It seems to me that we should not make a split between ideal point and point at infinity, because as far as I can tell they get used synonymously in each of the contexts. We simply have to describe the affine case and the hyperbolic case, and not define it in the general projective context. —Quondum 05:15, 18 June 2015 (UTC)

Thanks for rewriting, I think the term Omega point is used every now and then, I think it commes from a practice to use greek names for ideal points to distinguish them clearly from normal points (named P, Q and so on ). I was still thinking about splitting but you are more knowledgable than me :) WillemienH (talk) 08:42, 18 June 2015 (UTC)

Not necessarily, and in particular I do not have a broad literature base, so I'm weak on the actual terms in use. I rely on Google books etc. I may have some understanding of some of the actual concepts, even though I'm not sure of the terms. I cannot find the term "omega point" even though I think that it is a nice term, because "omega" is often used to mean "last" or "end", which to me works better than "infinity" because the concept applies even in the cases without a sense of distance, as with the general affine case. Perhaps you could put in a reference to its use and check whether its definition is the same? But anyway, we should relate it to the terms used in the article, where I've intuitively settled on "ideal point" rather than "point at infinity" for the same reason. The concept of an ideal point applies in the case of finite geometries too, where there is no ordering on points of a line (as with the complex case), and not even a concept of a neighbourhood (which the complex case retains). I usually jump in when a concept or context is presented as more specific than I believe it to be. The article could probably expand in the ideal points of a finite geometry. How would you feel about a rename of the article to "Ideal point"? —Quondum 13:21, 18 June 2015 (UTC)

"Omega point" ≡ "Ideal point" can be found in most elementary geometry texts that discuss non-Euclidean geometries (see especially the discussion of Omega triangles). They can be referred to as "points at infinity", but that term comes from a different tradition. As a finite geometer, I can emphatically say that "ideal point" is not used in the finite geometry context, they are always referred to as points at infinity. Bill Cherowitzo (talk) 17:33, 18 June 2015 (UTC)

Thanks, Bill. This means that we should not rename the article. I'm not entirely settled on the scope, and in particular my attempt to cover the use of a single definition to cover two fairly disjoint uses of the term "point at infinity". As such, would a separation into two articles make sense as suggested by WillemienH, one on the omega points of hyperbolic geometry? The concept would also apply in finite geometry, I'd think. —Quondum 19:01, 18 June 2015 (UTC)

Omega point is not much used and i would not support "Omega point (geometry)" as the main page. I think it would be better to use "ideal point" or "ideal point (hyperbolic geometry) ", but how often is idealpoint used in the other geometries? WillemienH (talk) 06:54, 19 June 2015 (UTC)

My impression was that close to half the books that I browsed via Google books used both "point at infinity" and "ideal point", often in statements like "The ideal points are called points at infinity", and that this was not absent in the hyperbolic case. But of course, my search terms could be skewing the proportions. I'll have to look more closely. If it turns out that "ideal point" is used more commonly than "point at infinity" in the hyperbolic case, your suggested "Ideal point (hyperbolic geometry)" would seem like a fair fit.

On a side note, I wonder how much the finite hyperbolic case has been studied. I looked into it a while back, and a projective geometry is not split into the hyperbolic interior, absolute and hyperideal (de Sitter, or is it anti-de Sitter?) spaces as with the real case, but it still similarly splits into three spaces. —Quondum 14:51, 19 June 2015 (UTC)

copied the section to ideal point as stub for that page (with hat notes and so added) will add more later (but feel freeto add. for your question about finite hyperbolic geometries http://www.employees.csbsju.edu/tsibley/Section-7.3.pdf claims there is an finite hyperbolic geometry of order 3 with 13 points. (it says a new version of this book is in preperation, should be published shortly WillemienH (talk) 08:29, 21 June 2015 (UTC)

I think that moving the hyperbolic case out is going to simplify things for both articles.

The claim says little about a general construction. I have a straightforward construction that I was using to generate a whole family of Desarguesian finite hyperbolic geometries in any number of dimensions, but it is a while since I looked at it; it might be interesting to kick it into life again. Since this is so simple, I'd expect it to be well-studied by now. —Quondum 15:18, 21 June 2015 (UTC)

End of history section WillemienH (talk) 22:02, 22 June 2015 (UTC)

the triangle is an ideal quadrilateral

?

Is it error?

May be the quadrilateral is an ideal quadrilateral? Jumpow (talk) 18:45, 9 February 2018 (UTC)

Yes. Fixed. Thanks for catching this. —David Eppstein (talk) 19:22, 9 February 2018 (UTC)

And beyond?

Presumably we should also have an article ultra-ideal point. Double sharp (talk) 10:32, 22 October 2018 (UTC)