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Revision as of 17:27, 23 June 2015

New start

Here is my contribution for a dedicated article: "Real projective line".Rgdboer (talk) 20:55, 21 June 2015 (UTC)

Thanks for great work! Tkuvho (talk) 07:32, 22 June 2015 (UTC)

non-desarguean?

The current version contains the footnote "If a real projective line happens to appear in a non-Desarguesian plane the harmonic structure cannot be presumed." Now the usual embedding of this would be in the real projective plane, rather than some other non-Desarguean plane. This is certainly possible for example by using the projective plane over the Cayley numbers. However, I am wondering about the relevance of this footnote at this page. Tkuvho (talk) 07:32, 22 June 2015 (UTC)

The embedding in a non-Desarguesian plane is but one possibility. If the usual axioms are applied to the line with no embedding, the group of motions can be considerably larger. So if the article is to mention the non-Desarguesian direction, this requires a more thorough covering the possibilities. —Quondum 13:24, 22 June 2015 (UTC)

The embedding of a real projective line in a non-Desarguean plane has no projective meaning, strictly speaking. Of course in differential geometry it does, but it is not the subject of this page. Tkuvho (talk) 14:37, 22 June 2015 (UTC)

I agree, that kind of embedding is not natural and mentioning non-Desarguesian planes seems quite out of place in this article. Bill Cherowitzo (talk) 17:39, 22 June 2015 (UTC)

Construction over the real line: can't be glossed over

The construction over the real line, recently added, needs considerable work. One cannot simply add a point to the real line as defined in that article and get the real projective line as a geometric object. That article gives a hopelessly unclearly defined object for the purpose. One needs to start specifically with the real affine line, a homogeneous space, whereas it is difficult, from that article, to think, geometrically, of anything other than an object of which the group of automorphisms is the trivial group. Secondly, even starting with a one-dimensional affine real space (also not Euclidean: it must have no metric structure), one has to include a construction for changing the added point into a normal point, i.e. expanding the group of motions in a particular way. So, placing this as the first alternative of a definition for the construction implies far too much assumption; this needs to be fixed. For now, I'd suggest removing this until a full section explaining the construction correctly can be added. —Quondum 13:44, 22 June 2015 (UTC)

The only problem is that all of the theorems of projective geometry such as Pappus, Desargues, Pascal, etc. become inapplicable to affine configurations if one does not include a mention of the construction via adding points at infinity, which is a considerable loss if one is interested in actual applications of projective geometry :-) Metric structures are irrelevant here. Tkuvho (talk) 14:39, 22 June 2015 (UTC)

Perhaps thinking of the construction of the real projective line as the set of all lines through the origin in R would be fruitful. Identifying these lines with their slopes gives the real line and the vertical line through the origin gives the point at infinity. The homogeneous nature of the result is quite apparent from this viewpoint. The non-uniqueness of the coordinatization (or embedding, if you prefer) can also be made very clear. Bill Cherowitzo (talk) 17:34, 22 June 2015 (UTC)

This view of the construction also shows that the first sentence in the definition section is not really correct. I would also advocate putting in a mention of the point at infinity, but this has to be done carefully. One should not start with the real line and think of adding a point to it, as this gets you into the difficulties that Quondum has mentioned (not insurmountable, but it requires some work to get it right). Rather, start with the real projective line and point out that the removal of one point leaves you with a copy of the real line and the removed point can be thought of as the "point at infinity" with respect to that copy. Bill Cherowitzo (talk) 16:56, 23 June 2015 (UTC)

Day-one review

Good participation for mid-summer. All comments directed to improved article with clear experience in bringing this topic out for general review. Some changes were made this afternoon reflecting discussion in the last 24 hours, including a link to Point at infinity showing that it is relative to chart selection. Please add to the See also as appropriate; my contribution is slope. This subtle little object from an old geometrical practice has an important place in math, we do well to explicate it clearly. As for the non-Desarguesian situation, that technicality that Hilbert and others used to upset expectations of old, it is mentioned as a caveat since it may arise in advanced studies.Rgdboer (talk) 23:25, 22 June 2015 (UTC)

Mathematics made difficult

This is probably the most complicated way I have ever seen to describe a circle. The lede says that the thing is homeomorphic to a circle, seemingly implying that it has a different geometric structure, and then gratuitously mentions that it is a non-trivial smooth manifold. Actually, though, the natural distance function on any real projective space is simply the angle between lines, which can be up to ${\displaystyle {\frac {\pi }{2}}}$. In this case, this is the same as the metric of a circle. So it's not just homeomorphic to a circle, it is a circle, specifically a circle of radius ${\displaystyle {\frac {1}{2}}}$. Will anybody mind if I simplify the lede accordingly? Also the construction itself is a little technical and is not intuitively explained - it should be stated explicitly that the points of ${\displaystyle \mathbb {R} P^{1}}$ are lines in ${\displaystyle \mathbb {R} ^{2}}$ passing through the origin. And we should definitely mention that it is commonly understood as the one-point compactification of the real line; even though, as Quondum notes, this does make it a little less clear what the metric is, we can do away with this problem by mentioning that the metric is that of the circle. (A little more technically, the Riemannian metric of the real-line-with-point-at-infinity model of ${\displaystyle \mathbb {R} P^{1}}$ is ${\displaystyle {\frac {dx}{1+x^{2}}}}$, which is obtained by pushing forward the Euclidean metric ${\displaystyle dx}$ by ${\displaystyle \arctan }$, the obvious map which takes points on the real line to angles.)

In the "automorphisms" section, it doesn't quite make sense to say "the mappings are homographies" - what we really mean is that the homographies are mappings of special interest. We also should probably mention that ${\displaystyle PGL_{2}(\mathbb {R} )}$ is the isometry group of the hyperbolic plane, which in the Poincare disc model can be seen as the interior of the circle. The real projective line then can be seen as imbedded in the complex projective line, and this explains the statement that the group of homographies is intermediate between the modular group and the full Moebius group. Does anybody mind if I make these changes as well?

I think the "Structure" section might do with some expansion - I really don't understand what it says and I'm not familiar with all that classical geometry stuff. --Sammy1339 (talk) 05:58, 23 June 2015 (UTC)

This comment expresses some legitimate concerns. There is one point I don't follow. It is true that the homographies are isometries of the hyperbolic plane, but why do you write "the real projective line then can be seen as imbedded in the complex projective line"? The imbedding in the complex projective line is merely the extension of coefficients, and needs not be explained in terms of hyperbolic isometries. Tkuvho (talk) 07:35, 23 June 2015 (UTC)

I guess that the homographies of the real projective line may be related to the motions of the omega points of the hyperbolic plane; this would explain the group being the same and may be worth a mention. This seems far more natural (or at least more intuitive) to me than the relationship with the complex projective line; Tkuvho's argument around on complexification makes sense. —Quondum 14:53, 23 June 2015 (UTC)

Well the idea is that you take ${\displaystyle \mathbb {C} ^{2}}$ and form the complex projective line, and look at what happened to the plane spanned by the two real axes: it became a real projective line which is the boundary of the upper half-plane ${\displaystyle H^{2}}$. The homographies of this line are precisely the Moebius transformations which fix it - these are the linear fractional transformations with real coefficients; they are also the isometries of ${\displaystyle H^{2}}$. This explains the relationship mentioned in the text, that the modular group (a group of hyperbolic isometries) is a subgroup of the group of homographies of ${\displaystyle \mathbb {R} P^{1}}$, which is in turn a subgroup of the Moebius group. --Sammy1339 (talk) 16:20, 23 June 2015 (UTC)

There is no connection whatsoever in what you describe between the representation of the real projective line as a subset of the complex projective line, on the one hand, and its representation as the "boundary" of the hyperbolic plane, on the other. Tkuvho (talk) 16:24, 23 June 2015 (UTC)

Of course there is - the hyperbolic plane (half-plane model) is half of the Riemann sphere a.k.a. ${\displaystyle \mathbb {C} P^{1}}$, and the ${\displaystyle \mathbb {R} P^{1}}$ constructed as above is the boundary of it. It's not a coincidence that the Moebius transformations (complex homographies) which fix ${\displaystyle H^{2}}$ (or equivalently which fix ${\displaystyle \partial H^{2}}$) are precisely the hyperbolic isometries. This occurs because the Moebius transformations are the conformal bijections which map circles to circles, and the geodesics of ${\displaystyle H^{2}}$ are circles which intersect the boundary orthogonally, so a transformation will be an isometry of ${\displaystyle H^{2}}$ iff it fixes ${\displaystyle \partial H^{2}}$, maps circles to circles, and preserves angles. --Sammy1339 (talk) 16:56, 23 June 2015 (UTC)

This is a misleading point of view. The complex homographies in general don't fix the real projective line, and form a much larger group, namely PSL(2,C), than the real homographies. Not much is gained by viewing the hyperbolic plane as "half" the complex projective line. Tkuvho (talk) 17:06, 23 June 2015 (UTC)

Right, so the point is that those elements of ${\displaystyle PSL_{2}(\mathbb {C} )}$ which do fix the real line are the real homographies. This is the way to find out that the isometry group of the hyperbolic plane is ${\displaystyle PSL_{2}(\mathbb {R} )}$ - I don't know of another way to prove this, offhand. It's also the view universally taken in the study of Riemann surfaces and analytic number theory. --Sammy1339 (talk) 17:19, 23 June 2015 (UTC)

But the hyperbolic metric has little to do with this, and it certainly cannot be induced by the imbedding into the complex projective line, so in a way using the phrase "hyperbolic plane" in relation to the complex projective line is neither here nor there. Fractional linear transformations act in the upperhalf plane and give all the isometries, so one doesn't need to ever speak of projective geometry to understand basic facts about the hyperbolic group of isometries. Of course the crossratio provides a nice way of defining the hyperbolic metric. Tkuvho (talk) 17:26, 23 June 2015 (UTC)