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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"? —
842:. 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.
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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.
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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
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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
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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.
355:"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
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443:", 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.
389:. 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
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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
749:
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;
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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.
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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
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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
745:
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
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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
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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
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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
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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
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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
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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
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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
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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
725:
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
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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
291:, 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
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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.
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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 :)
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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?
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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".
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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.
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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.
688:. 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. —
299:, 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. —
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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
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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
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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
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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
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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
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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).
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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. —
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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
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is premised on a confusion: between points points at infinity and omega points. I suspect that some authors may define the term
258:. 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.
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The superscript "2" makes printing the formulas fail; a superscript of "1" works correctly. I don't know how to fix this.
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the lead. However, I would be happy to start in the lead with a projective geometry with a privileged subspace. —
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dimensions if you start from an affine space (much simpler to see if you start from the projective space).
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The above should be linked here to illustrate the connection between R and RP^1, and also C and CP^1.
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I hope that the rewite of the lead has adequately addressed the concern about "the" expressed here. —
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Also i found the lead confusing is it about the one dimensional case or about the 2 dimensional case?
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Okay, fair point. Is there a general term in (projective) geometry for any one of the the sequence
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are quite extensively used in both the affine and hyperbolic families of geometries. I do not see
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Knowledge. If you would like to participate, please visit the project page, where you can join
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I think that moving the hyperbolic case out is going to simplify things for both articles.
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I still need to look at references to say anything definitive about the use of the terms
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of hyperbolic geometry? The concept would also apply in finite geometry, I'd think. —
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Since this is so simple, I'd expect it to be well-studied by now. —
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Okay, I've replaced "flat" with "projective space". —
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