Talk:Point at infinity

The superscript "2" makes printing the formulas fail; a superscript of "1" works correctly. I don't know how to fix this.

stereographic projection
The above should be linked here to illustrate the connection between R and RP^1, and also C and CP^1. Tkuvho (talk) 08:52, 8 April 2011 (UTC)

Hyperbolic geometry section
The section 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, . 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)

Not singular
We have line at infinity and plane at infinity which are both made up of points at infinity. Use of "The" in the lede to this article is improper. Rather the article should have a generic description of when a space (mathematics) is augmented by one or more points at infinity for some purpose.Rgdboer (talk) 22:44, 17 June 2015 (UTC)


 * Yes, this seems to be the case. It seems like the use is quite broad, including in hyperbolic geometry where every line has two such points.  The article will need to try to define the concept fairly generally.  —Quondum 23:24, 17 June 2015 (UTC)


 * I hope that the rewite of the lead has adequately addressed the concern about "the" expressed here. —Quondum 13:25, 18 June 2015 (UTC)

flats
The lede currently claims that the set of ideal points forms a flat, with a reference to the page on flats. If that's what's meant here, the comment is inaccurate because the flat article talks about Euclidean flats. Tkuvho (talk) 13:35, 18 June 2015 (UTC)
 * Agree. A flat is an affine subspace (or translate of such, depending on who's defining it) while the set of points at infinity form a projective subspace. The article Flat (geometry) is too specifically restricted to the Euclidean case and uses an unqualified term "space" when an affine space (or in the context of that article $R^{d}$) is intended. This just makes it harder to see the finer distinctions in terminology. Bill Cherowitzo (talk) 18:03, 18 June 2015 (UTC)
 * Okay, fair point. Is there a general term in (projective) geometry for any one of the the sequence point, line, plane, etc.?  —Quondum 18:47, 18 June 2015 (UTC)
 * Ya. Things, other things, still other things, etc. Seriously, from the synthetic point of view these are just the names of different types of primitive objects and are only "defined" by their incidence relations and whatever axioms are imposed on them. Having some type of general name implies a commonality which the theory, at this level of abstraction, does not support. Of course, if you restrict to the classical projective geometries (even up to allowing arbitrary skewfields) these are the traditional names given to the $d$-dimensional projective subspaces, which I suspect you already knew. Bill Cherowitzo  (talk) 05:52, 19 June 2015 (UTC)
 * Okay, I've replaced "flat" with "projective space". —Quondum 14:37, 19 June 2015 (UTC)

space v.s. plane
Thanks for the cleaning up. One question, though: why the restriction of the statement about a point per pencil to the plane? It is true in any dimension, and to limit the statement to the plane would only be for pedagogical reasons. —Quondum 00:14, 20 June 2015 (UTC)
 * Not quite. When you add a point per pencil, you are talking about a parallel pencil and this only exists in a plane. I think that that visualization is good for this article, so I changed "geometry" to "plane" to make it correct. The construction is a bit more complicated to explain in higher dimensions if you start from an affine space (much simpler to see if you start from the projective space). Bill Cherowitzo (talk) 03:08, 20 June 2015 (UTC)
 * You have me seriously puzzled. Yes, I am talking about a pencil of all lines parallel to each other, but this is defined in any number of dimensions, and I maintain the statement was correct.  I am not objecting to a visualization using the plane, but to state it that way in only two dimensions is suggestive of that being the only time that it applies, which we should avoid.  The word "geometry" may be non-ideal but "space" could be substituted.  —Quondum 04:22, 20 June 2015 (UTC)
 * This has to do with the definition of parallel lines. The definition for two lines in a plane (i.e., equal or do not intersect) does not extend to higher dimensions due to the existence of skew lines. The "plane definition" of parallel can be used for hyperplanes in any dimension. For other dimensional affine subspaces (such as lines) we say that two of the same dimension are parallel iff there exists a translation mapping one to the other. This is of course a more general concept as it applies to all types of affine subspaces, including points as well as hyperplanes. One of the things that is making this article a little awkward for me is that points at infinity are being described as extramural points added to the ends of lines instead of just being added to an affine space. The imagery is nice, especially in the plane case, but it leaves some significant open issues (such as how do we know what kind of structure these additional points form?) unresolved in higher dimensions. Bill Cherowitzo (talk) 20:56, 20 June 2015 (UTC)
 * So what if we use a different definition of what parallel lines are in a higher-dimensional space? They are still called parallel, we do not call skew lines in an affine space "parallel", and the concept of a pencil of parallel lines in a higher-dimensional space still applies. This is intuitive in the real case for any number of dimensions, especially three, where the points at infinity are the pairs of antipodes of the celestial sphere: ideal for the lead, no more difficult to imagine than the line at infinity in a plane, and it ties in with the origin of the word "projective".  Quibbles about the exact definition of "parallel" belong in the body, not as an argument to restrict a statement in the lead.
 * I agree that 'at the "end" of each line' is stretching things, and we should reword this first statement, which (amongst other problems) is a very bad fit for finite geometries. I was trying to capture both the affine and hyperbolic cases in a single "definition", but I'm uncomfortable with doing so; this is part of what underlies my support for a split of the article (even though the name gets reused, trying to treat them as the same topic is clunky).
 * Your last point, at least the parenthesized question, I thought was addressed by the statement "all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong". —Quondum 22:32, 20 June 2015 (UTC)
 * Let me try again. We do not use the term pencil in the way you want to use it. The only way to correctly use this term with your intended meaning is to restrict to the plane case. A pencil of lines is the set of lines through a point that lie in a plane. In general, a pencil of $n$-dimensional subspaces is the set of all the $n$-dimensional subspaces that contain a given $(n - 1)$-dimensional subspace and lie in a single $(n + 1)$-dimensional subspace. What you are trying to describe is a partial star of lines. A star of lines is the set of all lines through a point (in any dimension). Your intuitively obvious set of parallel lines is a star of lines through a point in the hyperplane at infinity without the lines through that point which lie in that hyperplane, restricted to the affine space obtained by the removal of that hyperplane (due to the restriction, you don't have to mention the lines that are being tossed − I just wanted to be perfectly clear as to what is going on). As to my parenthetical remark, yes the article does say that the points at infinity form a hyperplane, but what I was trying to get at was that this is a far from trivial leap when starting by adding points to lines. I am not advocating a need for a proof, just saying that this approach cries out for a proof to make that statement understandable. If you were to start by saying that the points at infinity are the elements of the hyperplane at infinity, this whole issue disappears. Bill Cherowitzo (talk) 03:50, 21 June 2015 (UTC)
 * Well, I guess that since most of the terminology is fairly new to me, and I tend to use it in ways that might not entirely fit with established use; we should obviously correct it to standard usage. The simplicity of starting with a full projective geometry is neater, but is a struggle for visualization – e.g., the images in Real projective plane are scant help.  For the layman, we could use the anchor of familiarity of an affine (Euclidean) geometry in the lead without the rigour, and derive that picture in the body starting from the projective space as you describe (yes, I know what is being tossed), showing that we end up with what's in the lead.  However, I would be happy to start in the lead with a projective geometry with a privileged subspace.  —Quondum 05:46, 21 June 2015 (UTC)
 * Those images really don't help, what we need is something that details the projectivization of an affine space. If I can't find an appropriate image, I'll make one. I can envision the lead as something like this ... "In projective geometry, affine spaces are turned into projective spaces by adjoining a hyperplane at infinity whose elements are called points at infinity. This process is reversible in the sense that if you start with a projective space, choose any hyperplane and remove the points of that hyperplane, the removed points will be the points at infinity of the resulting affine space. To an affine line only one point at infinity is adjoined, to an affine plane a line's worth of points at infinity are adjoined, to an affine 3-space, a plane's worth are adjoined, etc. Consider an affine plane, the process of adjoining points at infinity is done as follows: ... For affine spaces of dimension 3 or higher it might be easier to start with the projective space and ..." This is probably too much detail for the lead, but I think it gives the right flavor for this article. Bill Cherowitzo (talk) 21:06, 21 June 2015 (UTC)

"Improper point" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Improper_point&redirect=no Improper point] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at until a consensus is reached. Joy (talk) 19:05, 16 February 2023 (UTC)