![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Here is my contribution for a dedicated article: "Real projective line". Rgdboer ( talk) 20:55, 21 June 2015 (UTC)
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 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)
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)
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 . 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 . 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 are lines in 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 is , which is obtained by pushing forward the Euclidean metric by , 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 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)
Today the differential geometers' definitions were quoted to show their uniformity and presumption of a Euclidean sphere to make the definition. Meanwhile an editor has proposed a definition presuming knowledge of vector spaces and deriving equivalence from that structure. Surely mathematically minded people look at foundations of their concepts and we should do so here. The differential geometers are pragmatic but unconcerned about the projective structure that might result from fields besides R. Similarly, appealing to vector space knowledge may be distracting. That is why the original definition given two days ago (a purely algebraic construction) is to be preferred. Rgdboer ( talk) 23:21, 23 June 2015 (UTC)
The new lead asserts that the real projective line is a circle. This wrong. The real projective circle is homeomorphic to a circle, but it is not, as far as I know diffeomorphic to a circle. One may see that by observing that the universal cover of the projective line is not the projective line itself, but a projection of index two of a circle. Also, there is no embedding of the projective line into a Euclidean plane.
Moreover, the new lead does not have a neutral point of view, by privileging the topological point of view with respect to the other points of view (geometric, algebraic, differential, analytic, ...). For these reasons, I'll revert this change of the lead.
On the other hand, I am not satisfied by the previous state of the lead, and I will edit it, but only when the body of the article will be sufficiently expanded, for having a lead reflecting the body of the article. In fact the following sections are yet lacking: Projective frames and change of homogeneous coordinates, homographies, cross-ratio, ... D.Lazard ( talk) 16:12, 24 June 2015 (UTC)
Sorry I was wrong about everything I was saying about differentiable geometry: The projective line and the circle are diffeomorphic (that is isomorphic as differentiable manifolds) and even birationally equivalent as algebraic varieties (I believe, but I am not sure, that they are not isomorphic as real algebraic varieties). But this is not the question. Both standard definitions of the circle and the projective line make them naturally homogeneous spaces for some group, but not the same group. For the circle, this is the group of the plane rotations. For the projective line, this is the group of homographies PSL2(R), whose action is induced by the action of GL2(R) on the lines. Note that when the projective line is defined through synthetic geometry, one gets exactly the same homography group.
Thus, if the circle and the projective line are isomorphic as manifolds, they are very different mathematical objects. Beginning the lead by "the real projective line is a circle" is as ridiculous (and confusing for the layman) as beginning Parabola article by "the parabola is a line" ( Sammy1339's arguments apply exactly in the same way to this case). D.Lazard ( talk) 09:16, 25 June 2015 (UTC)
Isometric manifolds have all the same properties; they are the same manifold.I deduce that two different circles of the same radius in Euclidean plane are the same circle :-)
(Copied from preceding thread) Both standard definitions of the circle and the projective line make them naturally homogeneous spaces for some group, but not the same group. For the circle, this is the group of the plane rotations. For the projective line, this is the group of homographies PSL2(R), whose action is induced by the action of GL2(R) on the lines. Note that when the projective line is defined through synthetic geometry, one gets exactly the same homography group. D.Lazard ( talk) 09:16, 25 June 2015 (UTC)
Apparently, this discussion did not start from the article content, but from a side remark by myself in this talk page. For the moment, "homography" appears only as a single word in the lead, and in the last section. Synthetic geometry is not cited explicitly, but only implicitly, in the last section, through the words "central projections" and "parallel projections". This last paragraph, which I have never edited, says essentially the same as me, namely that classical geometry define the same homographies on the line as the modern presentation through linear algebra. IMHO this does not derserve to be discussed further, before having in the article a detailed definition/description of the homographies of the projective line. Then we could discuss how to describe the relationship between the classical definition and the modern one through linear algebra. D.Lazard ( talk) 14:44, 26 June 2015 (UTC)
Several comments. The circle and RP1 are diffeomorphic. They are also isomorphic as algebraic varieties. That does not mean that they are "the same". The circle is a particular conic section, for instance. It is a special case of an ellipse, for instance, any while every circle is isomorphic to every ellipse as an algebrair variety, it is not true that every ellipse is a circle. The real projective line, as the set of one dimensional linear subspaces of a two dimensional vector space over the reals, does not support a "natural" metric. It is the homogeneous space PGL (2)/B, where B is a borel subgroup. This is the Poisson boundary of the associated RiemannIan symmetric space PGL (2)/K, where K=SO (2) is the maximal compact subgroup (this is the hyperbolic plane). The real projective line naturally carries a projective structure, that is, a maximal atlas whose transition functiond are fractional linear. This can equivalwntly be given as a Cartan connection on a PGL (2) bundle over RP1 I. In contrast, the circle does carry an invariant Riemannian metric. Indeed, it is the unique one dimensional Riemannian symmetric space of compact type, which is a principle homogeneous space for the group SO(2). Sławomir Biały ( talk) 23:01, 26 June 2015 (UTC)
The link embedding of a real line in a projective line (on this change) seems incorrect to me: an embedding is an isomorphism, which requires all the structure of the embedded object to be present and to be preserved in the space in which it is embedded, unless the isomorphism is specified to be on a restricted structure. In particular, the the real line has a ring structure, whereas the image of the embedding does not (though it may be considered to unnaturally induce one). And even a geometric affine line has a problem in being embedded in a projective space in this sense: the groups of motions is different. Is there better a way of expressing this? It is really just an injective map, though there is clearly some structure that is preserved, such as the natural topology (which was why I selected that particular link, though it might not be a good one). — Quondum 03:23, 27 June 2015 (UTC)
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Here is my contribution for a dedicated article: "Real projective line". Rgdboer ( talk) 20:55, 21 June 2015 (UTC)
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 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)
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)
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 . 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 . 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 are lines in 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 is , which is obtained by pushing forward the Euclidean metric by , 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 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)
Today the differential geometers' definitions were quoted to show their uniformity and presumption of a Euclidean sphere to make the definition. Meanwhile an editor has proposed a definition presuming knowledge of vector spaces and deriving equivalence from that structure. Surely mathematically minded people look at foundations of their concepts and we should do so here. The differential geometers are pragmatic but unconcerned about the projective structure that might result from fields besides R. Similarly, appealing to vector space knowledge may be distracting. That is why the original definition given two days ago (a purely algebraic construction) is to be preferred. Rgdboer ( talk) 23:21, 23 June 2015 (UTC)
The new lead asserts that the real projective line is a circle. This wrong. The real projective circle is homeomorphic to a circle, but it is not, as far as I know diffeomorphic to a circle. One may see that by observing that the universal cover of the projective line is not the projective line itself, but a projection of index two of a circle. Also, there is no embedding of the projective line into a Euclidean plane.
Moreover, the new lead does not have a neutral point of view, by privileging the topological point of view with respect to the other points of view (geometric, algebraic, differential, analytic, ...). For these reasons, I'll revert this change of the lead.
On the other hand, I am not satisfied by the previous state of the lead, and I will edit it, but only when the body of the article will be sufficiently expanded, for having a lead reflecting the body of the article. In fact the following sections are yet lacking: Projective frames and change of homogeneous coordinates, homographies, cross-ratio, ... D.Lazard ( talk) 16:12, 24 June 2015 (UTC)
Sorry I was wrong about everything I was saying about differentiable geometry: The projective line and the circle are diffeomorphic (that is isomorphic as differentiable manifolds) and even birationally equivalent as algebraic varieties (I believe, but I am not sure, that they are not isomorphic as real algebraic varieties). But this is not the question. Both standard definitions of the circle and the projective line make them naturally homogeneous spaces for some group, but not the same group. For the circle, this is the group of the plane rotations. For the projective line, this is the group of homographies PSL2(R), whose action is induced by the action of GL2(R) on the lines. Note that when the projective line is defined through synthetic geometry, one gets exactly the same homography group.
Thus, if the circle and the projective line are isomorphic as manifolds, they are very different mathematical objects. Beginning the lead by "the real projective line is a circle" is as ridiculous (and confusing for the layman) as beginning Parabola article by "the parabola is a line" ( Sammy1339's arguments apply exactly in the same way to this case). D.Lazard ( talk) 09:16, 25 June 2015 (UTC)
Isometric manifolds have all the same properties; they are the same manifold.I deduce that two different circles of the same radius in Euclidean plane are the same circle :-)
(Copied from preceding thread) Both standard definitions of the circle and the projective line make them naturally homogeneous spaces for some group, but not the same group. For the circle, this is the group of the plane rotations. For the projective line, this is the group of homographies PSL2(R), whose action is induced by the action of GL2(R) on the lines. Note that when the projective line is defined through synthetic geometry, one gets exactly the same homography group. D.Lazard ( talk) 09:16, 25 June 2015 (UTC)
Apparently, this discussion did not start from the article content, but from a side remark by myself in this talk page. For the moment, "homography" appears only as a single word in the lead, and in the last section. Synthetic geometry is not cited explicitly, but only implicitly, in the last section, through the words "central projections" and "parallel projections". This last paragraph, which I have never edited, says essentially the same as me, namely that classical geometry define the same homographies on the line as the modern presentation through linear algebra. IMHO this does not derserve to be discussed further, before having in the article a detailed definition/description of the homographies of the projective line. Then we could discuss how to describe the relationship between the classical definition and the modern one through linear algebra. D.Lazard ( talk) 14:44, 26 June 2015 (UTC)
Several comments. The circle and RP1 are diffeomorphic. They are also isomorphic as algebraic varieties. That does not mean that they are "the same". The circle is a particular conic section, for instance. It is a special case of an ellipse, for instance, any while every circle is isomorphic to every ellipse as an algebrair variety, it is not true that every ellipse is a circle. The real projective line, as the set of one dimensional linear subspaces of a two dimensional vector space over the reals, does not support a "natural" metric. It is the homogeneous space PGL (2)/B, where B is a borel subgroup. This is the Poisson boundary of the associated RiemannIan symmetric space PGL (2)/K, where K=SO (2) is the maximal compact subgroup (this is the hyperbolic plane). The real projective line naturally carries a projective structure, that is, a maximal atlas whose transition functiond are fractional linear. This can equivalwntly be given as a Cartan connection on a PGL (2) bundle over RP1 I. In contrast, the circle does carry an invariant Riemannian metric. Indeed, it is the unique one dimensional Riemannian symmetric space of compact type, which is a principle homogeneous space for the group SO(2). Sławomir Biały ( talk) 23:01, 26 June 2015 (UTC)
The link embedding of a real line in a projective line (on this change) seems incorrect to me: an embedding is an isomorphism, which requires all the structure of the embedded object to be present and to be preserved in the space in which it is embedded, unless the isomorphism is specified to be on a restricted structure. In particular, the the real line has a ring structure, whereas the image of the embedding does not (though it may be considered to unnaturally induce one). And even a geometric affine line has a problem in being embedded in a projective space in this sense: the groups of motions is different. Is there better a way of expressing this? It is really just an injective map, though there is clearly some structure that is preserved, such as the natural topology (which was why I selected that particular link, though it might not be a good one). — Quondum 03:23, 27 June 2015 (UTC)