Talk:Hyperbolic geometry/GA1

GA Review

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Reviewer: Sammy1339 (talk · contribs) 14:38, 13 August 2015 (UTC)[reply]

The article has severe deficiencies in scope that will require very substantial changes to fix. Nowhere is the notion of hyperbolic distance mentioned, or any of the nice formulas such as area of a triangle = sum of its angles. I'd like to see geodesics in the hyperbolic plane explained, and for someone to tell me that a hyperbolic circle is a circle. There ought to be a picture of a right-angled hexagon.

The topic of the article is not exactly clear - I'm gathering it's really about hyperbolic plane geometry as it doesn't discuss higher dimensions or hyperbolic surfaces. Very problematically, nowhere is "hyperbolic" clearly defined. The article should better explain what "constant negative curvature" means. The statement that hyperbolic plane geometry is the geometry of "saddle surfaces" is strange and a bit misleading - while the embedding into euclidean 3-space makes it easier for laypeople to visualize, this embedding is never used in the rest of the article and serves largely to create confusion when the article goes on to talk about different models.

Several statements in the article are unclear, for example the connection to Minkowski space. I guess that the connection is that a mass shell in phase space is a hyperboloid model of hyperbolic 3-space, and knowing this would allow one to add rapidities. Some of the details could be explained.

The final section, "Homogeneous structure," is not very readable and lacks context. The notion of a hyperbolic isometry should be plainly laid out, and the action of the automorphism group PSL(2,R) should be explained, perhaps with explicit examples. While I don't advocate dumbing things down, readers who understand the notion of a "Riemannian symmetric space of noncompact type" are not the target audience for this article, so such notions need to be explained, or at least expanded upon, as they are introduced.

✗ Fail --Sammy1339 (talk) 14:34, 13 August 2015 (UTC)[reply]

Thanks for this review. I agree with many of its criticisms and it encapsulates well some of the things that I was finding myself uncomfortable with in the article, which were causing me to hold off on doing a review myself. (Also, now that someone else has done a review, I no longer feel that I should delay making improvements to the article to avoid disqualifying myself as a reviewer.) —David Eppstein (talk) 17:30, 13 August 2015 (UTC)[reply]

Start of thanks for review by willemienH

Thanks for your review, gives lots of things to work on. The problem is a bit it is quite a complex subject and I want to make it understandable to the interested lay person, my idea was to make hyperbolic geometry the introduction to the subject while the more complex points could go to hyperbolic space. I do find the points you raise almost all fair, but to give a point by point comment:

the notion of hyperbolic distance, correct needs explaining It is just the length of the shortest path, geodesic between the points.

Well, there are formulas, though. In the half-plane model, if you define $u(z,w)={\frac {|z-w|^{2}}{4\Im {z}\Im {w}}}$ then $d(z,w)=arccosh(1+2u(z,w))$ . --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

that is only about the Poincare half-plane model therefore I would not add it in this article.WillemienH (talk) 09:49, 15 August 2015 (UTC)[reply]

the nice formulas such as area of a triangle = sum of its angles No that is not a valid formula in hyperbolic geometry, the valid formula is mentioned see hyperbolic geometry#Standardized Gaussian curvature "The area of a triangle is equal to its angle defect in radians."

Very stupid of me, sorry. Should be \pi minus the sum of angles, as you said. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

Don't forget to multiply by radius of negative curvature squared JFB80 (talk) 20:56, 25 January 2016 (UTC)[reply]

I'd like to see geodesics in the hyperbolic plane explained again crazy enough it is just a straight line.

No, in the half plane and disc models, they are lines or circles which intersect the boundary orthogonally. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

that is only about the Poincare half-plane model and Poincare disk model therefore I would not add it in this article. WillemienH (talk) 09:49, 15 August 2015 (UTC)[reply]

for someone to tell me that a hyperbolic circle is a circle: but a circle is a circle, there is nothing more to it. (exept that its circumference and area differ)

The set of points equidistant from a point z in the hyperbolic half plane is a Euclidean circle, but centered at a point lying somewhere above z. That's a non-trivial fact. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

that is only about the Poincare half-plane model therefore I would not add it in this article.(last time) WillemienH (talk) 09:49, 15 August 2015 (UTC)[reply]

There ought to be a picture of a right-angled hexagon good idea, but there is nowhere (yet) a good spot to put it. (the article only mentions triangles, nowhere it mentions other polygons. mayby when I find a right-angled hexagon on a saddle surface.

Better to show it in the disk model - and yes, the current lack of a place to put it is the real problem. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

The topic of the article is not exactly clear - I'm gathering it's really about hyperbolic plane geometry as it doesn't discuss higher dimensions. Fair point I will try to add something to the lead, but it is correct that the article is mainly about hyperbolic plane geometry. (I wanted to keep it simple, for higher dimensions there is hyperbolic space.

It doesn't discuss hyperbolic surfaces not sure what you mean by this.

Compact Riemann surfaces of genus at least two admit a hyperbolic metric. They can be created by quotienting the hyperbolic plane by the action of a discrete group of hyperbolic isometries. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

I don't find the hyperbolic metric something inportant again it is (in my eyes)mostly belonging to some models

Very problematically, nowhere is "hyperbolic" clearly defined. Fair point, but the problem is it doesn't really mean anything, it is just the name given to this geometry by Felix Klein.

As I understand it, hyperbolic means constant negative scalar curvature. Although I might just be making this up: Benedetti and Petronio use a more restrictive definition, which is equivalent to saying that a hyperbolic manifold is one that is locally isometric to some hyperbolic space. In two dimensions it's the same thing, anyway. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

Oh, never mind. The Killing-Hopf theorem says that these two definitions are equivalent. --Sammy1339 (talk) 21:58, 13 August 2015 (UTC)[reply]

The article should better explain what "constant negative curvature" means. Fair point, but it is a bit a mix between "Gaussian curvature", "constant curvature" and "negative curvature"

The statement that hyperbolic plane geometry is the geometry of "saddle surfaces" is strange and a bit misleading - while the embedding into euclidean 3-space makes it easier for laypeople to visualize, this embedding is never used in the rest of the article and serves largely to create confusion when the article goes on to talk about different models. It is not misleading hyperbolic plane geometry is the geometry of "saddle surfaces" with a constant curvature. I think this is how Riemann saw it. But I agree it needs more explaining, but unfortunedly this becomes very quickly very complicated. (a saddle surface with a constant curvature is nothing more than a part of an "pseudospherical surface of the hyperbolic type" but then there is not yet an article on pseudospherical surfaces, let alone that it mentions the different types)

Maybe an idea to make a section on "the geometry of saddle surfaces" where all the points about saddle surfaces are described and explained.

I think the whole bit about saddle surfaces should be no more than a footnote. We should focus on the half-plane, disk, and maybe hyperboloid models. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

A saddle point for negative curvature only occurs in a Euclidean representation and the only plane example of it is the pseudosphere. Hyperbolic space is importantly represented on a hyperboloid in Minkowski space where there is no saddle point. On the subject of saddle points there is duplication in the two articles Hyperbolic plane geometry as the geometry of saddle surfaces and Physical realizations of the hyperbolic plane JFB80 (talk) 21:23, 25 January 2016 (UTC)[reply]

Several statements in the article are unclear, for example the connection to Minkowski space. : Fair point, I find Minkowski geometry quite unclear myself. (but do you mean this in the section hyperbolic geometry#Geometry of the universe (Special relativity) or in the section hyperbolic geometry#the hyperboloid model

The connections to relativity could be clarified better, in both places. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

I guess that the connection is that a mass shell in phase space is a hyperboloid model of hyperbolic 3-space, and knowing this would allow one to add rapidities. Some of the details could be explained. I am not sure about this myself.

The final section, "Homogeneous structure," is not very readable and lacks context. The notion of a hyperbolic isometry should be plainly laid out, and the action of the automorphism group PSL(2,R) should be explained, perhaps with explicit examples. While I don't advocate dumbing things down, readers who understand the notion of a "Riemannian symmetric space of noncompact type" are not the target audience for this article, so such notions need to be explained, or at least expanded upon, as they are introduced: Fair points again, I don't understand this section myself. was thinking about moving it to hyperbolic space (that is a bit where i want to move every ting complicated to).

That makes sense. This topic is a good candidate for an audience split, like introduction to general relativity. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

Thanks for your review it give lots to work on. (But a bit sad that you demoted it from B-class to C-Class on the quality scale, I thought the article improved since my major edits in Januari 2015.) Still thanks for you work I hope you can comment on the things I found unclear and other comments you my have. WillemienH (talk) 21:00, 13 August 2015 (UTC)[reply]

I appreciate that, and I don't mean to insult your work. The topic is a fairly big one and writing a comprehensive article on it will be a large task. Explicitly restricting the scope might help - such as by splitting to introduction to hyperbolic geometry and hyperbolic plane. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)[reply]

I think the 13th we were almost editing this post at the same time :), I added a bit to the lead addressing the name and that in this article is mainly about the 2 dimensional case. I disagree to add formula's to the article that are only about one particular model of geometry (but feel free to add them to the Poincare half-plane model), I want the article general. the halfplane model is just one of the 4 models of hyperbolic geometry and then we could add the related formula's of the other models and so on. I will add more later, I was thinking about a new section on "geometry of saddle shaped surfaces",to adress the points about curvature and saddle points, but first extend the "properties" section to address the points you made on circles , distances and lines (and to add a link to Absolute geometry in the mean time) why was that one missing.WillemienH (talk) 09:49, 15 August 2015 (UTC)[reply]

I strongly disagree with your claim that distance formulas are unimportant. Yes, they are model-specific, so they should be given in more than one model, but without them there is no basis for any kind of numerical calculation in this geometry. By analogy, the (Pythagorean) formula for distances in Euclidean geometry is equally important (it's the single displayed formula in Euclidean geometry), and also equally dependent on a specific model (Cartesian coordinates). The fact that it depends on the Cartesian coordinate system is not a good reason for ignoring it. Same here. —David Eppstein (talk) 17:52, 15 August 2015 (UTC)[reply]

I agree with the addition from User:JRSpriggs about coordinate distance [1] because it is general and aplies to all models. (although it could be more specific that it is about Lobaschevski coordinates and so, see we need an article on hyperbolic coordinate systems Martin's "foundations of geometry and the non-euclidian plane" has a nice chapter with i thought 4 different hyperbolic coordinate systems and that just for the 2 dimensional case) , see this article just keeps growing :) WillemienH (talk) 08:48, 30 August 2015 (UTC)[reply]

Regarding coordinate systems for the hyperbolic plane. One can define coordinate systems for the hyperbolic plane by pulling the standard coordinate systems used with Euclidean geometry (Cartesian and radial) back through the maps which take the hyperbolic plane to its Euclidean models. It is also possible to define coordinate systems directly in the hyperbolic plane using the geometric constructions available there as was done at Hyperbolic geometry#Distance. This one is not associated with any of the standard models, but does give a one-to-one correspondence between the points of the hyperbolic plane and elements of ℝ×ℝ.

I noticed that the straight lines have equations of the form

x=C

or

\tanh y=A\cosh x+B\sinh x\quad {\text{ when }}\quad A^{2}<1+B^{2}

where A, B, and C are real parameters which characterize the straight line.

Is it worthwhile to include this in the article? JRSpriggs (talk) 12:07, 3 September 2015 (UTC)[reply]

I am not sure, maybe it would better in a new article for hyperbolic plane coordinates where we could describe all kinds of hyperbolic coordinates systems. (Martin mentions 4.) But until that page is started, I think it will be okay. I was wondering what are the formulas of lines in the hyperboloid model and would like to have that at that page. WillemienH (talk) 18:19, 4 September 2015 (UTC)[reply]