# Talk:Hyperbolic geometry

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## MacFarlane

A fourth model is the Alexander MacFarlane model, which employs an 3-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 4-dimensional euclidean space. This model is sometimes ascribed to Karl Weierstrass. Macfarlane used hyperbolic quaternions to describe it in 1900.

I snipped the above from the article and replaced with something more accurate. I've never heard of this model being called this name and I can't even find a single reference on Google (except for wikipedia-related hits). Not to mention, MacFarlane is not the originator of this model, as far as I know. For example, John Stillwell credits Poincare; some credit Killing and say Poincare generalized it. I've never heard it being attributed to Weierstrass. I guess the part on hyperbolic quaternions may be ok, so I left it in. --C S 12:45, Dec 16, 2004 (UTC)

If Alexander MacFarlane's contribution to the Royal Society at Edinburgh was the first published description of the hyperboloid model, should he not be so credited ? Rgdboer 00:44, 2 Apr 2005 (UTC)
Sure, why not; but to avoid further controversy, it would be best to have a reference to the MacFarlane paper and/or to the secondary source that makes this claim. linas 02:44, 2 Apr 2005 (UTC)
Renolds reference is earliest suggestion so far Rgdboer 00:09, 7 January 2006 (UTC)
Nice job! --C S (Talk) 03:25, 7 January 2006 (UTC)

## Hyperbolic plane vs. hyperbolic geometry

I have often wanted to separate off the hyperbolic plane stuff to its own page. By hyperbolic plane I mean the unique simply connected two-dimensional surface with constant curvature −1. Hyperbolic geometry is a much more general term than just this. For one thing there are many hyperbolic Riemann surfaces that aren't simply connected (most in fact). Hyperbolic geometry should also refer to higher dimenisonal manifolds with similiar geometry (such as hyperbolic 3-space).

I started writing a draft of the hyperbolic plane article about six months ago (see User:Fropuff/Draft 2) but soon abandoned it and moved on to other things. I wasn't sure how to properly organize things. There are 4 main models of the hyperbolic plane:

1. The Minkowski model (which is usually taken as the definition)
2. The Klein model
3. The two Poincaré models (or conformal models)

My question was whether or not these should all be discussed in the same page or if we should have separate pages for each. If separate pages, should the Poincaré models be discussed on the same page or separate pages? I think at least the Minkowski model should be the same page as the hyperbolic plane page and taken as the definition. Anyone have any thoughts on the matter? -- Fropuff 17:04, 2005 Mar 10 (UTC)

I've written separate articles for all of the above, and then linked them to the article on hyperbolic space; however, there is still not a separate article on the hyperbolic plane. I think defining hyperbolic space as the hyperbolic model/Minkowski model is a bad idea, since there are various models. Moreover, the more fundamental model underlying both the hyperbolic and Klein models is the projective semialgebraic model, which defines a distance function on a projective semialgebraic variety; from this we get both the hyperbolic and Klein models by normalizing. I've put a discussion of this and the various models derivable from it in the hyperbolic space article. Gene Ward Smith 21:33, 19 April 2006 (UTC)
You asked for an opinion :) ... There should be an introductory article (such as this one) that gives a one or two paragraph into to each of the models, and mentions things like multiply connnected surfaces, etc. However, one can say a lot about the poincare models alone, and so the paragraphs should link off to the full-detail article on the poincare model.
Unfortunately, I am the one to blame for the current disconnected state of the Poincare model related articles. For a while I had one article with included both the Poincare metrics for both plane and disk, the symmetries for the plane and disk, the schwarz-alhfors-pick theorem for the plane and disk, and it just got so long that splitting it up seemed to be the right thing to do. So I split it up ... but now it just feels disconnected and scattered. I was going to continue cleaning up and integrating and shuffling the contents so that a nicer table of contents resulted, but never quite got around to it.
Anyway, I have vague plans for continuing with the series, e.g. adding teichmuller spaces, and maybe in the process I'll clean up the 2d hyperbolic stuff a bit more. I know very very little about 3d & 4d hyperbolic things. (I gather no one does, outside of ed witten) linas 18:43, 20 Mar 2005 (UTC)

I'm not opposed to having separate pages for the two Poincaré models, although having a third page for the Poincaré metric seems overly redundant. Each model of the hyperbolic plane should discuss its own form of the metric.

As far as higher dimensions go, we need to have a separate page for hyperbolic space Hn (which currently redirects here). Witten may know more the average guy about hyperbolic manifolds, but he is certainly not the expert. Entire books have been written about hyperbolic 3-manifolds. -- Fropuff 02:06, 2005 Mar 22 (UTC)

I was joking about Ed Witten. He's just mindblowing to listen to when he talks. Although 'tHooft is arguably even more fun. You'll notice I added a reference for a book by matsuzaki and tanaguchi for the 3-manifold case (its in the article Kleinian group). As the book deals primarily with 3-manifolds and secondarily with kleinian groups, feel free to copy the reference .. here ?linas 15:58, 24 Mar 2005 (UTC)
Oh,and just to be clear, I think the poincare half-plane and disk should be treated together in one article. Essentially all of the theorems and facts for one apply to the other; the're so closely mirrored it would be strange to separate them. I could try to merge the metric article back in with some other article. Not sure which. Low priority for me right now. I'll be reviewing Riemann surfaces shortly, and writing some articles about that, so maybe a clearer structure will emerge once I immersse my self in that a bit. linas 16:05, 24 Mar 2005 (UTC)
Oh, and one more comment: Klein model should probably redirect to Kleinian group and have one article discuss both. If/when that article bloats into something large, it could be split into two. But, for now, I think it would make sense to keep them together. linas 16:10, 24 Mar 2005 (UTC)

Is this type of structure dealt with adequately here? As a non-mathematician I noticed it seldom discussed in articles about hyperbolic plane surfaces. epinet.anu.edu.au/mathematics/p_surface Mydogtrouble (talk) 19:28, 8 February 2010 (UTC)

## Escher circle limit III

I am removing the parenthetical statement about geodesics in Escher's circle limit III. It read:

The famous circle limit III and IV [2] drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can clearly see the geodesics (in III they appear explicitly).

the reason for this is that the white lines in CLIII are not geodesics. The angles on the triangles are slightly less than 60 degrees and the angles on the squares (mmm ... equilateral equiangular quadrilaterals) are slightly more than 60 degrees. If they were geodesics they would meet the "bounding circle" at right angles. They clearly don't. Do a google search on "geodesics in circle limit III" and the link http://www.ajur.uni.edu/v3n4/Potter%20and%20Ribando%20pp%2021-28.pdf comes in near the top, and it explains it. Andrew Kepert 09:45, 23 Jun 2005 (UTC)

## Hjelmslev transformation

# The fifth model is the Hjelmslev transformation. This model is able to represent an entire hyperbolic plane within a finite circle. This model, however, must exist on the same plane which it maps, and therefore non-Euclidean rules still apply to it.

This appears to have been written by someone unfamiliar with the other models of hyperbolic geometry, as both the projective disk and conformal disk models have the properties stated. In addition, I've never heard of a fifth model called the Hjelmslev transformation (it would be strange to call a model a transformation, in any case), although there is another model called the upper hemisphere model that I've been meaning to add. Also, when I look at Hjelmslev transformation, it just describes the conformal disk model (also called Klein model), so it would seem to be redundant. My literature search makes me suspect that the Hjelmslev transformation refers to a map between two of the known models. In addition, the only references of Hjelmslev I could find were in reference to Hjelmslev planes and axioms which appear to be a more general setting than that of hyperbolic geometry. ---C S (Talk) 08:41, 17 January 2006 (UTC)

Uh... yeah, sorry. I added that entry on Hjelmslev's transformation, and I wrote what is at Hjelmslev transformation so far. I cannot say that I know much about the Klein model, except that I got the impression that the disk was, in fact, Euclidiean. The Hjelmslev transformation may be a step between the the actual plane and the tidy Klein model, but it is also an independent model of it's own. A geometer needn't only do geometry in an Euclidean plane to be certain of his results. I remind you that Lobachevsky did his whole work on hyperbolic geometry without the aid of models. If one were accustomed to working in this geometry, the Hjelmslev transformation could be seen as a tremendous time-saver and proof-simplifier. Proofs about parallel and ultra-parallel lines become very clear as a result of this transformation. Since I do not understand the rules which govern the Klein model, and no article has been created explaining it, I think it is hasty to remove this passage about this transformation. All I know is that this is a legitimate model of hyperbolic geometry. If it belongs under some other heading, I think there ought to be an article created for the sake of this. And the explanation of exactly how these other models work. Oh, I don't know. --- SJCstudent 07:23, 18 January 2006 (UTC)
I'm on something of a wiki-break, so I really shouldn't be editing this :-) but I thought you should get a prompt response. I don't understand your statement about the Klein model being "Euclidean". It isn't, as all the "models" of hyperbolic geometry are in fact respresenting hyperbolic geometry. If you mean that geodesics in the Klein model look like straight chords in the disc, your Hjelmslev model has that same feature from your pictures and description. In fact, your pictures of parallel, ultra-parallel, etc. look like the standard pictures of parallel, ultra-parallel, etc. in the Klein model! Not to mention that the properties you state at the end of Hjelmslev transformation:
1. The image of a circle sharing the center of the transformation will be a circle about this same center.
2. As a result, the images of all the right angles with one side passing through the center will be right angles.
3. Any angle with the center of the transformation as its vertex will be preserved.
4. The image of any straight line will be a finite straight line segment.
5. Likewise, the point order is maintained throughout a transformation, i.e. if B is between A and C, the image of B will be between the image of A and the image of C.
6. The image of a rectilinear angle is a rectilinear angle.
also hold for the Klein model. So I'm afraid you haven't said anything to clarify why this model is different than the Klein model. In addition, I can't find any mention of the Hjelmslev model in any of my books on hyperbolic geometry. The only mention I find on the Internet is of something seemingly more general. Also, my search of published mathematical papers by Hjelmslev doesn't turn up a model of hyperbolic geometry. His papers on infinitesimal and projective geometry, as I said, appears to be of a more general foundational nature.
To allay your fears, let me mention that yes, I am familiar that geometers work in spaces more general than a Euclidean plane and I am aware Lobachevsky did not use a model. Despite all this, I have not heard of the Hjelmslev tranformation, nor do I know of some model of hyperbolic geometry by Hjelmslev that looks so much like the Klein model. I request that you cite a source you are using to create your Hjelmslev transformation article. As it stands, it looks like a ripe candidate to be merged/moved into Klein model with correct historical attributions. --C S (Talk) 22:37, 23 January 2006 (UTC)
Ok, here is the deal. 1. The difference I am trying to highlight is this: the Klein Model places an infinite hyperbolic plane within a finite euclidean circle, the Hjelmslev transformation places an infinite hyperbolic plane within a finite Lobachevskian circle. The lines in the example circles look straight because they are straight, I have preserved the image of straightness in the finite diagrams. Either way... 2. I am looking for some textual basis for this transformation outside of my non-euclidean college textbook. I have written the author of the manual and spoken to other professors who are in charge of the department. I will either provide you with proper evidence soon, or alter my college's curriculum. Either way, I will have something soon. Thank you for your patience. SJCstudent 17:15, 25 January 2006 (UTC)
If I understand the article correctly, it is equivalent to what you would get by taking the Klein model and shrinking it by some scaling factor k; that is, each vector ||u||<1 in the Klein model becomes ku, where 0<k<1. In terms of the hyperbolic space, we have a model of hyperbolic space in a finite ball in hyperbolic space. As k-->0, this approaches the Klein model since the curvature of the region covered by the ball goes to zero. Gene Ward Smith 22:26, 20 April 2006 (UTC)
FWIW, I have seen older books (books concentrating on ruler-compass constructions, as opposed to "modern" algebraic constructions) that have a model of the hyperbolic plane with the geodesics being straight (angles are of course not preserved). I just don't remember the name of that "model". linas 01:05, 26 January 2006 (UTC)
That model is called the Klein-Beltrami model or the projective disk model. SJCstudent claims to be talking about something different. But what he is talking about I'm not quite sure yet. -- Fropuff 01:41, 26 January 2006 (UTC)

Ok, I did some extensive research. But, before I reveal what I have found, I think there are a few misunderstandings I should attempt to clarify first. One: for the last time... the Klein model projects an hyperbolic plane into a EUCLIDIAN circle, the Hjelmslev transformation projects an hyperbolic plane into an HYPERBOLIC circle. I do not know how I can make the distinction more clear than this. Two: I am not denying that these two models are very similar in alot of ways, this does not however make one more primary. For example... I could have easily said that the Klein model is just some ripoff of the Hjelmslev transformation. Either way....

The 16th volume of the mathematical series "International Series of Monographs in Pure and Applied Mathematics" is entitled "Non-Euclidean Geometry" and is written by Stefan Kulczyscki. It was trasnlated from Polish by Dr. Stanslaw Knapowski of the University of Pozan. Copywright 1961 by Panstwowe Wydawnictwo Naukowe Warszawa. It was originally prinited in Poland. Its Library of Congress Card Number is 60-14187.

In this book, sections 9 and 10 detail the creation and use of the Hjelmslev transformation. Please respond asap. I feel that this outside textual source justifies the reinsertion of the transformation into the article. SJCstudent 18:39, 3 February 2006 (UTC)

H.S.M. Coxeter writes in his review of that book (in the American Mathematical Monthly Vol. 69 No. 9 p. 937 available through JSTOR:)

The mathematical development begins with Hjelmslev's theorem (see e.g. Coxeter, Introduction to geometry, Wiley, NY, pp. 47, 269), which enables the author (following Hjelmslev himself) to prove that a particular transformation of hyperbolic space ("mapping j") is a collineation. O being a fixed point, each ...[Coxeter explains the Hjelmslev transformation]...The whole space is thus transformed into the interior of a sphere whose chords represent whole lines. We thus have the Beltrami-Klein projective model imbedded in the hyperbolic space itself!

From the start, I thought it was something like this, a transformation, rather than a new different model of hyperbolic geometry. It should be noted that maps between (and into) different models are not uncommon, and I am familiar with several of these. For example, the Klein model is often times embedded into the projective plane (historically, this was one way it was discovered by Klein), and can consequently show up in the visual sphere of an observer in a higher dimensional hyperbolic space (as viewed through the upper half space model). The Lorentz model also offers different ways to view the Klein and Poincare disc models inside the Lorentz space.
My conclusion is that despite your reference (Coxeter mentions it is a good book by the way), it does not justify listing Hjelmslev transformation as a separate model of hyperbolic geometry. It certainly deserves some mention but perhaps under "see also" or some other explanatory section. --C S (Talk) 22:17, 3 February 2006 (UTC)
This seems satisfying. I apologize for not fully recognizing the distinction between "model" and "transformation" earlier. A "See also" section is suiting... Sorry for all the hassle. SJCstudent 01:31, 4 February 2006 (UTC)

For all the talk about the hyperbolic plane and even hyperbolic space (where people might generally be thinking of hyperbolic space of more than two dimensions, although I know hyperbolic 1-space is an example of hyperbolic space), hyperbolic 1-space, the "maximally symmetric, simply connected [of course any connected 1-dimensional manifold, even the circle, is simply connected]," 1-dimensional "Riemannian manifold with constant sectional curvature −1," is not often talked about, and I am curious about it. What is it called? The hyperbolic line? I can tell that hyperbolas are not examples of hyperbolic-1 space as they are not connected and clearly do not have constant curvature. How many dimensions of Euclidean space does it take to isometrically embed hyperbolic 1-space? Or what I'm really looking for is: how many dimensions of Euclidean space does it take for hyperbolic 1-space to be embedded in and be as much itself, if you know what I mean, as the circle is in the Euclidean plane and the [i]n[/i]-sphere is in Euclidean [i]n[/i]+1-space? That may be eqivilent in all cases to a manifold being able to be isometrically embedded in a certain space, but I'm not sure. It seems like that number of dimensions must be greater than two, because a curve having constant nonzero curvature and being confined to the Euclidean plane would seem to have to be a circle. But would three Euclidean dimensions be enough for hyperbolic 1-space to "naturally" fit? The number of dimensions it takes to isometrically embed hyperbolic 1-space could shed some insight into the number of dimensions it takes to isometrically embed hyperbolic 2-space, which I believe has been narrowed down to 4 or 5 now but I'm not sure if it's been proven that it doesn't take 6 dimensions. Any answers to these questions would be greatly appreciated. Kevin Lamoreau 05:20, 4 June 2006 (UTC)

I'm overwhelmed by the full question, but I don't think it makes sense to talk of negative curvature of a 1-dimensional surface. Guassian curvature is intrinsic property as the product of the principle curvatures, and curvature defined as the reciprical of the radius of tangent circles. The sign is only meaningful if there are 2 or more dimensions to the surface. So you can ask for a 1-dimensional surface of constant curvatures either +1 OR -1, but both are simple unit circles embedded in a Euclidean 2-space or higher. I guess that doesn't help much on your question. Tom Ruen 17:57, 4 June 2006 (UTC)
See my reply at Talk:Hyperbolic space. -- Fropuff 18:17, 4 June 2006 (UTC)
Thanks for both of your replies, Tomruen and Fropuff. I've looked at the curvature article and I get what both of you are saying. If it wasn't for the absolute value function, the curvature of the circle would be positive if the circle were defined parametrically c(t) = (x(t),y(t)) with the circle going counter-clockwise as t increases, and negative the circle were difined parametrically with the circle going clockwise as t increases. I also, after some pondering (I can have a thick head sometimes), get Fropuff's point in Talk:Hyperbolic space about all one-dimensional Reimannian manifolds being locally isometric, not just locally homeomorphic. So both of your replies were helpful. Thanks again, Kevin Lamoreau 20:02, 5 June 2006 (UTC)

It seems we are going back and forth on the inclusion of Systolic geometry in the "See Also" section. The link is somewhat relevant (and I think "spam" is a little harsh here), as there is mention on the target page of the use of systolic ideas in a hyperbolic setting. But the See Also section is getting pretty darn long, so it makes sense to open a discussion of how many links there should be, and which ones should be included. If it is too long, and some links should be pruned, then Systolic geometry would probably be one to go. After all, many topics in mathematics have as much connection to hyperbolic geometry as that does. Comments? -- Spireguy 02:18, 26 April 2007 (UTC)

A lot of links! This is an important topic, and the article should be expanded, so that these links can be incorporated into the text. Geometry guy 22:08, 24 June 2007 (UTC)

## "small scale" relative to what?

The following sentence is in the article:

"On small scales, therefore, an observer would have a hard time determining whether he is in the Euclidean or the hyperbolic plane."

At first reading, this didn't sit well with me. In Euclidean geometry, it doesn't matter at what scale you're working, and I don't think there's a way to specify the scale becaule the plane is infinite. (Does this make sense to anyone else?)

Then I figured that these "small scales" in hyperbolic geometry may be relative to the curvature of the hyperbolic space, or something like that. It seems that the more 'curved' the space is, the farther you have to 'zoom in' (and the smaller distances you have to consider) before the angle of parallelism approaches 90°. Given a hyperbolic space, does it have a parameter that describes its curvature? Might it be useful to add a few words on this topic to the article? Oliphaunt (talk) 12:55, 15 May 2008 (UTC)

In the hyperbolic plane, the angle defect of a triangle is the product of the curvature and the area of the triangle. In particular, it is directly proportional to the area. Hence, the smaller the area, the smaller the angular defect. Symbolically, by the Gauss-Bonnet theorem
${\displaystyle KA=\theta _{1}+\theta _{2}+\theta _{3}-\pi }$
where K is the curvature, A the area of the triangle, and the θi are the angles of the triangle. silly rabbit (talk) 13:31, 15 May 2008 (UTC)
OK, but do I then understand correctly that a hyperbolic plane is characterised by its curvature K? Shouldn't this concept then be more prominently discussed (or linked to) in the article? Oliphaunt (talk) 14:07, 15 May 2008 (UTC)
Yes, that's right. In fact a hyperbolic plane is the unique connected simply-connected complete Riemannian manifold of constant negative curvature K < 0. (Similar statements hold in higher dimensions, but with constant negative sectional curvature instead. See Space form.) This fact is rather important, and probably deserves to be mentioned somewhere. silly rabbit (talk) 14:27, 15 May 2008 (UTC)
As a result, each hyperbolic plane has a length scale which is ${\displaystyle {\frac {1}{\sqrt {-K}}}}$ where ${\displaystyle K\!}$ is the (constant) curvature of the plane. If the diameter of a region is small compared to that, then it will be approximately Euclidean. JRSpriggs (talk) 20:49, 15 May 2008 (UTC)
Thanks for the extra info. It would be nice if this could be added into the article somehow. Oliphaunt (talk) 21:20, 15 May 2008 (UTC)

## Exponential growth

The first is that the link relating to exponential growth points to the wrong page: it points to a page on a rather technical group theory topic rather than to the page on exponential growth.

My other point was silly. It didn't even dawn on me that it was referring to the hyperbolic circumference divided by the hyperbolic radius. Sorry 'bout that. --Hurkyl (talk) 20:53, 10 February 2009 (UTC)

On the hyperbolic plane, the circumference of a circle is
${\displaystyle C(r)=2\pi \sinh(r)\,}$
assuming that the curvature is -1. Integrating to get the area gives
${\displaystyle A(R)=\int _{0}^{R}C(r)dr=2\pi (\cosh(R)-1)\approx \pi \exp(R)\,.}$
Clear? JRSpriggs (talk) 09:40, 11 February 2009 (UTC)

## Gyrovector spaces - please remove

The section on gyrovector spaces is very poorly written, and there doesn't seem to be any evidence that this theory yields new insight into hyperbolic geometry. As far as I can tell from the linked article, someone has recently stumbled upon a less convenient formulation of symmetric spaces (a very well-established theory), and proposed to explain sundry mysteries of the universe (e.g., the nature of dark matter) by some ill-defined means. It looks like some combination of crackpot and original research which, as I understand, is frowned upon here. —Preceding unsigned comment added by 18.87.1.150 (talk) 21:30, 15 February 2010 (UTC)

Perhaps the section should be reduced to a sentence. The main article, Gyrovector spaces, is now well-developed with several significant sources. One of them, Scott Walter (2002), indicates the vital subject of relativity and modern mechanics in hyperbolic geometry. Walter credits the creator of gyrotheory with cultivating this vital thread in science. But his mention of an "elegant non-associative algebraic formalism" reminds me of hyperbolic quaternions; non-associativity is seldom elegant. Furthermore, Walter says the gyrotheory is most useful for comparing "elegance of respective proofs of selected theorems".Rgdboer (talk) 02:34, 16 February 2010 (UTC)
I've rewritten the section. I think it should stay. Charvest (talk) 07:36, 17 February 2010 (UTC)
What, there's a formalism for measuring elegance? —Tamfang (talk) 19:06, 29 September 2010 (UTC)

The difference between hyperbolic vectors and euclidean vectors is that addition of vectors in hyperbolic geometry is nonassociative and noncommutative.

Could this be reworded in a way that doesn't contradict the standard axiomatic definition of a vector (Euclidean or otherwise)? Dependent Variable (talk) 04:28, 28 September 2010 (UTC)
No. A vector is something with magnitude and direction. Something with magnitude and direction doesn't have to be compatible with the axiomatic definition of a vector space. Relativistic velocity addition is not associative or commutative (except when the velocities are in the same direction). Therefore relativistic velocities demonstrate that the axiomatic notion of vector space does not encompass all types of vectors. 84.13.72.71 (talk) 08:11, 28 September 2010 (UTC)
How about replacing the word "vector" with "gyrovector" here (given that a vector space is a special case of a gyrovector space, rather than the other way around, according to Gyrovector space)? A nonassociative or noncommutative vector is just a contradiction in terms.
"Something with magnitude and direction" is an appeal to intuition used in elementary texts, rather than an actual definition. The axioms give a precise definition which accounts for how the word is actually used in mathematical literature. A car has a direction and a magnitude, but that doesn't make it a vector. A polynomial, a color (in the context of a color space), a linear transformation, an infinite sequence of real or complex numbers, a discrete probability distribution can all be characterised as vectors, with appropriate definitions of addition and scalar multiplication. A vector space doesn't have to have a norm (defining magnitude) or an inner product (defining angles and hence a notion of directions).
If physicists call relativistic velocities "vectors", it could be for historical reasons (because the concept descends from that of velocity in Newtonian mechanics), and because they're represented as triples of real numbers (such objects being widely treated in other contexts as vectors). For some authors, they may be defined as vectors in terms of some overarching structure, such as spacetime algebra (a kind of geometric/Clifford algebra). It is often pointed out that relativistic velocity is not a vector associated with Minkowski space, only the spatial components of one in a particular, arbitrary basis. Dependent Variable (talk) 11:42, 7 October 2010 (UTC)
I've ammended the article to say gyrovectors rather than vectors are noncommutative. Of course in newtonian mechanics both position and velocity can be represented by the same type of vector and geometry, whereas in relativity position is unbounded but velocity is bounded so they have different geometries and notions of vector (n-tuples of R).
And yes it is the right way round. Every vector space is a gyrovector space. The axioms for gyrovectors are less strict as they don't require full associativity. So gyrovectors encompass a larger class of objects. Hyperbolic space can be seen as deformed euclidean space. Euclidean space is just a gyrovector space with a zero amount of deformation. 78.148.177.200 (talk) 13:54, 7 October 2010 (UTC)
It would be better to represent Lorentz boosts in Minkowski space by multiplication of matrices rather than addition of gyro-vectors. JRSpriggs (talk) 13:19, 28 September 2010 (UTC)
Yes. Gyrovector addition does not represent boosts. "Gyrovector addition" just means 3D-velocity composition. A boost is a transformation, between two different frames of reference, of the 4D-spacetime coordinates of events. Boost composition is associative. 3D-velocity composition is not associative. 84.13.64.158 (talk) 19:41, 1 October 2010 (UTC)
What is velocity composition if it is not following one boost by another? JRSpriggs (talk) 20:40, 1 October 2010 (UTC)
Velocities are closed under composition - a velocity followed by a velocity results in a velocity.
Boosts are not closed. A boost in one direction followed by a boost in a different direction is not just a boost in a new direction, but a boost plus a rotation. 84.13.208.35 (talk) 10:31, 2 October 2010 (UTC)

(unindent) If you are in spaceship O, and there is a spaceship A with velocity v relative to you, and a spaceship B with velocity u relative to A, then you can ask: "What is the velocity of B relative to you?". That's velocity composition.

Suppose the two velocities are not in a line but in different directions, then:

The coordinate transformation between O and A is a boost,
the coordinate transformation between A and B is a boost.
"What is the transformation between O and B ?". That's boost composition.
The answer is that the coordinate transformation between O and B is not a boost - if the axes at A are aligned with O and the axes at B are aligned with A, then the axes of B will be rotated relative to O. (Thomas precession). 84.13.66.230 (talk) 12:00, 2 October 2010 (UTC)

Thank you for the link to Thomas precession. I agree that the composition of two boosts is frequently not a pure boost, but includes rotation. I will have to think some more about what this means for velocities. JRSpriggs (talk) 12:27, 3 October 2010 (UTC)
The details of the relationship between boost composition and velocity composition have now been added to Velocity-addition formula#Velocity composition paradox. 89.241.232.46 (talk) 18:23, 1 November 2010 (UTC)
Yes but it's wrong because boost (short for Lorentz boost) is a matrix. The velocity that it acts on is the Minkowski 4-velocity so you should be talking about composing 4-velocities not boost matrices. Actually all the troubles originate from of the attempt to combine Cartesian vectors supplemented by the artificial gyro rotation. Gyrovectors are not needed because velocity is actually a vector in hyperbolic space as in the theory of Varicak and for such vectors there is a natural rotational effect when they are combined caused by parallel transport in curved space. Already in 1913 Borel remarked on this rotation which was later confirmed with the Thomas precession (see the article of Scott-Walker cited in the Varicak article).JFB80 (talk) 21:31, 5 December 2010 (UTC)
This is getting really off-topic for this page which is the talk page for hyperbolic geometry. However, yes it's true that a boost is usually represented as a 4x4 matrix. The boost matrix B(v) means the boost B that uses the components of v, v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used in the section Lorentz transformation#Matrix form. (To digress slightly: The formulae in the matrix entries depends on the way spacetime coordinates are represented. That article and most standard texts represent spacetime as (ct,x,y,z) and putting c=1 this becomes (t,x,y,z), but you could just as easily use (t,x,y,z) to start with (in which c is not used here not even c=1) and then you'd have to change the matrix entries to work with (t,x,y,z) instead of (ct,x,y,z), but that is a digression.) The point is that the matrix entries depend on the components of the 3-velocity v, and that's what the notation B(v) means. You could argue that the entries depend on the components of the 4-velocity because 3 of the entries of the 4-velocity are the same as the entries of the 3-velocity, but the usefulness of parameterizing the boost by 3-velocity is that the resultant boost you get from the composition of two boosts uses the components of the 3-velocity composition u${\displaystyle \oplus }$v in the 4x4 matrix B(u${\displaystyle \oplus }$v). But the resultant boost also needs to be multiplied by a rotation matrix because boost compostion (i.e. the multiplication of two 4x4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4x4 matrix that corresponds to the rotation gyr[u,v] to get B(u${\displaystyle \oplus }$v) = B(u${\displaystyle \oplus }$v)gyr[u,v] = gyr[u,v]B(v${\displaystyle \oplus }$u). Admittedly the notation gyr[u,v] was previously used as the rotation applied to a 3-velocity, whereas here the same notation is used to mean the rotation of spacetime coordinates, and this has caused confusion, but it is at root talking about the same rotation: namely the rotation that arises from the composition of two boosts. The notation gyr in context refers to the rotation, whether the rotation of 3-velocities or the rotation of 4-coordinates, it refers to the rotation. I've added this explanation to the velocity-addition article to clear up the confusion. 2.97.18.221 (talk) 08:37, 7 December 2010 (UTC)
I dont think you understood what I said which is very relevant to hyperbolic geometry particularly the subject of parallel transport which I believe offers the correct explanation of what are called gyrovectors. See particularly the diagram parallel transport.png.JFB80 (talk) 17:17, 7 December 2010 (UTC)
In your previous comment you say "Gyrovectors are not needed because velocity is actually a vector in hyperbolic space", but gyrovectors are vectors in hyperbolic space. You also say "as in the theory of Varicak", but Varicak in his 1924 book abandoned attempts to formulate a vector algebra in hyperbolic geometry because he couldn't get vector algebra to work, which Scott Walter points out in the "concluding remarks" of the article you mention.
As you point out, parallel transport of a vector around a closed loop back to the starting point in curved space leaves the final vector pointing in a different direction than the starting vector. For relativistic velocities, this is Thomas precession - the "natural rotation" you talk of. The rotation makes, as you put it, "the attempt to combine Cartesian vectors", nonassociative and noncommutative (I assume this is what you are referring to by "troubles"), but the whole point of the gyrovector approach is to have a Cartesian vector algebra for hyperbolic geometry. 89.241.225.113 (talk) 20:20, 7 December 2010 (UTC)
I've changed the articles to use upper case Gyr for the 4-spacetime rotation and lowercase gyr for 3-space rotation. I had mistakenly used gyr for both. 2.97.24.14 (talk) 10:30, 9 December 2010 (UTC)

(1) You said: 'gyrovectors are vectors in hyperbolic space'. That is just an assertion, it needs demonstration and proof (which should have appeared in the article on gyro-vectors.) I have not seen a proof and dont believe it is possible to give one. . (2) What S Walter (1999) said was: 'adapting ordinary vector algebra for use in hyperbolic space was just not feasible, as Varicak himself had to admit'. which is not quite what you said and is also not what Varićak (1924) said which was 'In Lobachevsky space, in contrast to Euclidean space, there is a large variability in the form of geometrical constructions. For this reason, one encounters difficulties in transferring theorems from the usual vector algebra to Lobachevsky space' (Quoted verbatim from Kracklauer's 2006 translation). Varićak did formulate a vector algebra after discussing these difficulties. (3) Thomas precession occurs during motion in a path with continuous tangent. Parallel transport of vectors round paths with discontinuous tangents e.g. triangles, gives the sort of behaviour ascribed to gyrovectors. On a spherical analogy, compare the diagram I quoted (parallel transport.png) with a corresponding diagram in Sommerfeld's 1909 Phys. Z paper (Wikisource) (4) You said 'but the whole point of the gyrovector approach is to have a Cartesian vector algebra for hyperbolic geometry. The quotations (2) are saying that is not possible, which is also my opinion. If you maintain it is possible you should prove it.JFB80 (talk) 22:07, 9 December 2010 (UTC)

You're right. The Gyrovector space article should explain the link between a gyrovector space and a hyperbolic space and not just assert that they are the same. It should show that angles, lines, and distances in a gyrovector space are the same as in the model of hyperbolic geometry that the gyrovector space is supposed to represent; that a gyrovector which is a pair of points and the geodesic between those points are the same as hyperbolic geodesic linking two points. So that gyrolines, gyroangles and gyrometric match the hyperbolic lines, angle and metric. Ungar's books do show that the line element, curvature, geodesics etc are the same. There is also some info in this paper. I'll see what else I can find amongst the gyrovector publications and in due course I will add such info to the article. In 1999 Walter may have said "just not feasible", but later in 2002 Walter gave the review: "Ungar furnishes the crucial missing element from the panoply of the non-Euclidean style: an elegant nonassociative algebraic formalism". I'm not sure which Sommerfeld paper you are referring to. Could you provide a link ? 89.241.232.203 (talk) 00:28, 10 December 2010 (UTC)
I am glad you agree about the necessity of proof. Yes, Walker expresses different opinions in his two articles. The important point is that Varićak was misquoted. Refer to Fig.2 in the Sommerfeld paper. The link is::[[1]]JFB80 (talk) 19:23, 10 December 2010 (UTC)

## Four models of hyperbolic geometry

Are 4 models of hyperbolic geometry equivalent to each other? How do you show that by maths? Thank you! Milk Coffee (talk) 14:09, 28 October 2010 (UTC)

For each natural number n ≥ 2, there is, up to isomorphism a scale transformation, only one simply connected hyperbolic space of dimension n without a boundary. Thus the four models (the Beltrami–Klein model, the Poincaré disk model, the Poincaré half-plane model, and the hyperboloid model) all describe the same structure for hyperbolic geometry. Where they differ is in how they represent that structure within Eucidean space. In some cases, the articles on those models contain mappings between the model being described in the article and some of the other models. See: Beltrami–Klein model#Relation to the hyperboloid model, Beltrami–Klein model#Relation to the Poincaré disk model, and Poincaré disk model#Relation to the hyperboloid model. Unfortunately, we (at Wikipedia) do not have a formula for conversion between the half-plane model and any of the others (although I am sure that someone knows how to do this). JRSpriggs (talk) 23:39, 28 October 2010 (UTC)
Conversion between disc and upper-half models: Hyperbolic Voronoi diagrams made easy page 6. 89.241.235.146 (talk) 12:05, 1 November 2010 (UTC)
This may be a rather late reply, but may help any new reader trying to get an understanding: The main difference between the 4 models is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic space. The characteristic feature of the hyperbolic space itself is that it has a constant Ricci scalar curvature of -1. For a given metric space, the matrix representation of the metric tensor, the Ricci curvature tensor, etc. can be different for usage of different coordinate charts (i.e. the different models). However what is characteristic to the metric space and is indifferent to the coordinate chart used, is the Ricci scalar curvature. So, using any of the models, if you compute the Ricci scalar, you'll find that the values match. In short, Ricci scalar is an invariant of Riemannian manifolds. Another similar invariant of metric spaces are the geodesics (i.e. geodesics map to geodesics under coordinate transformation) and hence studying their intersections tell us a lot about the metric space itself. Hyperbolic geometry generally is introduced in terms of this later type of invariant. Once we define a coordinate chart (one of the many possible "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the Ricci curvature of Euclidean space is 0). There is nothing very special about these 4 models - there are infinitely many more. However the embedding in Euclidean space due to these 4 specific charts show some interesting characteristics.- Subh83 (talk) 07:29, 14 March 2011 (UTC)
Thanks again! Milk Coffee (talk) 15:43, 14 March 2011 (UTC)
If there is popular interest, an abridged version of this explanation may be added in the main article for better understanding. Reference can be found in "Introduction to Hyperbolic Geometry, A. Ramsay" - Subh83 (talk | contribs) 23:04, 17 March 2011 (UTC)
By combining JRSpriggs' discussion and mine above, I am adding a brief subsection under the 4 models section. Feel free to improve. - Subh83 (talk | contribs) 17:37, 18 March 2011 (UTC)

Can Somebody PLEASE just provide an example of this?? Which models? Reference! I would like to read on this.

"Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid." 79.131.121.133 (talk) —Preceding undated comment added 01:57, 2 February 2012 (UTC).

The ones described in the "Models of the hyperbolic plane" section of this article. —David Eppstein (talk) 03:11, 2 February 2012 (UTC)
Especially the Beltrami-Klein model where the relation between Euclidean and hyperbolic planes is quite obvious. JFB80 (talk) 21:37, 2 February 2012 (UTC)

In A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces each of the 5 well known models is treated individually up to dimension n and are (conformally) unified together with Spherical AND Euclidean spaces, new theorems are also deduced.Selfstudier (talk) 10:44, 16 February 2012 (UTC)

## Suggestion to reorganise

Intro
- History
- Main characteristic properties of Hyperbolic Geometry (new heading)
- - non intersection lines (existing section)
- - Triangles(existing section)
- - Circles, disks, spheres and balls (existing section)
- Models of the hyperbolic plane(existing section)
- - Other models of the hyperbolic plane(existing section)
- - Connection between the models(existing section)
- Visualizing hyperbolic geometry
(Advanced characteristic properties of Hyperbolic Geometry ? )
- Homogeneous structure
rest


I would like to add again https://en.wikipedia.org/w/index.php?title=Hyperbolic_geometry&curid=241291&diff=644565668&oldid=642445150 ( by https://en.wikipedia.org/wiki/User:Peter_Buch )but they do need referencing.

This will be I think a major structural change so therefore please comment. WillemienH (talk) 10:47, 30 January 2015 (UTC)

• A new section Properties has been formed as you suggest. Thank you for moving some references in-line. Buch's open circle is something interesting but no reference is known so far. The History section needs links to mathematical structure and model (mathematical logic) so that the general reader can understand how the speculative structure became accepted. One of our challenges in such a broad article is that spheres and balls are discussed in Properties before any Model of the hyperbolic plane. It is important that an interested reader could gain enough understanding from each paragraph to proceed reasonably to the next.Rgdboer (talk) 00:49, 31 January 2015 (UTC)
• Buch's "open circles" are more commonly known as hypercycles, and the curvature formula can be found (in an equivalent form) e.g. here (2nd full paragraph on p.20; see MR2357448 for publication data). So I don't think that material is problematic. —David Eppstein (talk) 01:21, 31 January 2015 (UTC)

I started with a rewrite , moved history up and rewrote introduction. added links to saddle surface and sectional curvature (was thinking to remove info over ultra parallel and triangles triangles in the introduction but will wait till after rewrite properties)

Was thinking about a section special curves under properties but that is all for later WillemienH (talk) 02:00, 31 January 2015 (UTC)

If Peter Buch or someone else wants to talk about horocycles or hypercycles, then he should refer to them as such rather than calling them "circles" which they are not. JRSpriggs (talk) 07:02, 31 January 2015 (UTC)
Who cares what some editor wants to call them? The point is that the formula for the curvature of a circle is known, not original research. —David Eppstein (talk) 07:10, 31 January 2015 (UTC)
It matters because it is confusing to use the wrong words. I was confused by what he said. A circle is the locus of points at a certain distance from a given point. That is not the same as the locus of points at a certain distance from a given line (on one side of that line) which is what a hypercycle is. Nor is it the same as the object which is the limiting case between circles and hypercycles, i.e. a horocycle.
There is also the question of how does one define "curvature" in the context of hyperbolic geometry? JRSpriggs (talk) 07:23, 31 January 2015 (UTC)
I imagine as it is defined in geodesic curvature. —David Eppstein (talk) 07:50, 31 January 2015 (UTC)

I rewrote large parts of the properties section , but kept the old sections, (marked with old at the end of the title) I think the old sections can be removed because they don't contain useful information , but would like to hear the opinion from others on this. (also i think that under the triangle subsection info is that should be moved to hyperbolic triangle where it is lacking. WillemienH (talk) 17:51, 31 January 2015 (UTC)

## Relation between Euclidean and hyperbolic geometry

Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.

Can someone please clarify this in the lead? How can something "obey the axioms of hyperbolic geometry", such as that there are at least two lines through point P that do not intersect line R, while being "within Euclidean geometry"?

Thanks. Loraof (talk) 20:37, 11 April 2015 (UTC)

The models map lines of hyperbolic geometry to other curves (i.e. non-lines) in the Euclidean model, or otherwise to intersections of planes with a surface (or suchlike) when embedded in a Euclidean geometry of higher dimension. Because the lines do not correspond to those of Euclidean geometry, there is no contradiction. This gets tricky to explain briefly in the lead, so I'm not sure I'll be able to address it. —Quondum 21:38, 11 April 2015 (UTC)
To repeat what Quondum said in more concrete terms, consider the Klein model whose (hyperbolic) points are the interior points of a fixed disk and whose (hyperbolic) lines are the open chords of that disk. These objects are perfectly normal Euclidean objects that live within Euclidean geometry and so obey all the axioms that define that geometry. In the model however these are interpreted as hyperbolic points and lines and one can use the Euclidean properties of these objects to show that the (interpreted) hyperbolic axioms are satisfied. Geometers have described this situation as "relative consistency" since the model can be used to translate any contradiction that may exist in hyperbolic geometry into one that would have to exist in Euclidean geometry (and have pointedly ignored what the logicians have to say about consistency.) Bill Cherowitzo (talk) 04:56, 12 April 2015 (UTC)

I was thinking about removing the whole passage, it is to complicated at the point where it is . Allready under history it is mentioned that "In 1868, Eugenio Beltrami provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was." and further on there is a section on "Models have been constructed within Euclidean geometry " so the whole passage doesn't really add any thing. WillemienH (talk) 10:21, 12 April 2015 (UTC)

But that also made me think about the next paragraph maybe "Because each of Euclidean geometry and hyperbolic geometry is consistent, and both have similar, small sectional curvatures, an observer will have a hard time determining whether his environment is Euclidean or hyperbolic. Thus we cannot decide whether our world is Euclidean or hyperbolic."(old version 12/04/2015) Is better replaced by something like

"Because both Euclidean geometry and hyperbolic geometry are consistent, and in the physical world we cannot physically construct lines long enough to decide if they will eventually meet or not we cannot decide if space is curved or not and with that if the geometry of our world is Euclidean or hyperbolic. we can only decide that the curvature of space is higher than (latest measure of space curvature with ref) "

There are some problems with this passage , what are those lines really? elastic string, to short; light rays , bend under gravity; gravity lines don't seem straight either and so on. (the last calculation of curvature I know off was made when it was still believed that light rays were straight and there are people who disagree with the used method) WillemienH (talk) 10:21, 12 April 2015 (UTC)

I'm all for removing the mention of models as a proof of consistency from the lead, and keeping it in the body where it can be explained better. Also, the lead does need a rewrite. Your suggested sentence could be trimmed ("When the curvature is low enough, the distinction between hyperbolic, Euclidean and elliptic geometries becomes locally immeasurable, a problem that we have in determining the large-scale geometry of the universe.") or it could be removed from the lead too. —Quondum 16:26, 12 April 2015 (UTC)
I do agree that what I wrote above as draft needs a bit rewriting (that is why I wrote it here) but your sentence makes it to simple. it is not just a question on determing the curvature, it is also about how to measure the curvature in the first place. and without knowing the curvature hyperbolic geometry becomes rather meaningless, curvature is implicit in every formula on hyperbolic geometry, also I do think a link to sphere-world needs to be included. Also I think a bit of it needs to be included in the lead but it is all rather complicated, both geometries assume "homogenity of space, each point is similar " and "curvature is constant " while maybe in the real world that is not the case (and both geometries are incorrect) WillemienH (talk) 00:17, 13 April 2015 (UTC)

## Geometry of the universe

Hyperbolic geometry § Geometry of the universe does not give me the sense of really "belonging" in this article. The relevance of hyperbolic geometry in special relativity is that all subluminal velocities form a hyperbolic space, which is unrelated to the geometry of the universe. Describing our universe as three-dimensional (by thinking of it as Euclidean, elliptic or hyperbolic) is a throwback to Newtonian/Galilean thinking that time and space are in some sense separable. Since as Minkowski geometry displaces Galilean geometry (= 3-d Euclidean + time of Galilean relativity) in special relativity, if one wishes to discuss deviations from flat space (Minkowski geometry), rather than the Euclidean/hyperbolic/elliptic geometry discussion, the appropriate geometries to consider are Minkowski space/anti-de Sitter space/de Sitter space. —Quondum 16:45, 3 May 2015 (UTC)

Hi Quondom, I split Hyperbolic geometry # Geometry of the universe off from Hyperbolic geometry # Philosophical consequences because it was a seperate subject (physical instead of philosophical). I do think it belongs here but, but I am wondering if this section goes in the right direction, this section should (as i see it) be about the space part of space time alone. Space time is a 4-manifold and the question is what is the curvature of the 3-manifold that is space part of it. I do think it needs rewriting but I am not knowledgable enough, I do think that you are right in your idea that it is to be able to decide if the geometry of the universe is that of Minkowski space , anti-de Sitter space and de Sitter space. as you mentioned. WillemienH (talk) 18:08, 3 May 2015 (UTC)
Hyperbolic geometry serves as an easily-imagined analogy for exploring the question of a global geometry of the universe, and possibly served as a trigger or at least a confirmatory parallel to encourage thinking in this direction, but this is supposition on my part. However, trying to define a "space part" is problematic in the general case, though I suppose under an assumption of isotropy, the comoving frame would do, in which case the question as posed in the article might make sense (suitably reworded to reflect this choice, and might even be how cosmologists think of it, for all I know). In any event, some research is needed to determine historical relevance and influence. I am no expert. —Quondum 19:38, 3 May 2015 (UTC)
The question is more basic than you describe, it is not about analogy but really about the question "Is the geometry of space Euclidean, hyperbolic or spherical?"
Maybe this simplified description of the experiment helps:
One of the concequences of hyperbolic geometry is that the sum of angles of a triangle is less than 180 degrees this even applies to giant triangles earth -Sun - very distant object X. This means that the sum of the two angles not at the distant object X always add to less than 180 degrees. The experiment was to measure and add these angles. Unfortunedly when this was measured and calculated some very close value to 180 degrees was measured, far inside the tolerance of the instruments. meaning that there could be no other conclusion than: "We don't know if the geometry of space is euclidean, hyperbolic or sphereical, but if the geometry of space is hyperbolic the curvature is very close to zero, and the absolute length is a very long measure." I hope this explains it a bit (see also http://math.stackexchange.com/q/954483/88985 ) WillemienH (talk) 06:36, 4 May 2015 (UTC)
My point is that this question implicitly violates the principle of equivalence, by completely ignoring it and assuming a privileged frame (the comoving frame that I mentioned). If you try to measure the angles of a triangle moving relative to the sun, you might find that your results change significantly. One first has to demonstrate that the question is frame independent – and one will find that it is not. In our universe, the geometry of the universe looks different in different inertial frames, which tells you that it is a bad question to ask. The reason is that the geometry of space is four-dimensional, and trying to ask things about the geometry of an arbitrary 3-d section of it normally would make no sense. It is like saying: what is the shape of the section through a cube? There are other considerations that exacerbate this. There are 4-d geometries that a locally indistinguishable from a Minkowski space, but are surprisingly different from the four-dimensional global structures we have in mind as possibilities: they are topologically different from something like "hyperbolic space + time". The whole topic is simply too wide open to be using in an article about a geometry that does not consider time as part of the geometry. —Quondum 14:32, 4 May 2015 (UTC)
Not sure about your moving triangles at all, I thought the whole idea was "independent of time, devoid of mass" [1] (so there is no triangle moving in time or relative to the sun, the sun is just one of its (real) vertices. Also could you give some references to where your ideas lead to? (also notice that this measure predated general and special relativity ) WillemienH (talk) 11:55, 5 May 2015 (UTC)

References

Under a heading like "Geometry of the universe", if the text is referring to the history of thought prior to special and general relativity, I think it that this should be made abundantly clear in the wording. In my wording above ("principle of equivalence", "Minkowski space") it should be clear that I'm thinking in the context of relativity. —Quondum 15:48, 5 May 2015 (UTC)
I think you are right and did split the "Geometry of the universe" in a "predating relativity" and a "including relativity" part, (not the nicest titles but I could not think of anything better at the moment) For the "including relativity" part I did pick bits of what you wrote above, please check and enlarge where needed,(some references would be nice) thanks for the discussion. It did teach me that Minkowski space has some euclidean dimensions, and that the anti-de Sitter space is (in my humble and not scientific opinion) the real thing WillemienH (talk) 18:53, 5 May 2015 (UTC)
Yes, better, though it will need some care not to have OR. I'll think about the wording. Interestingly, any (entirely space-like) section of a de Sitter space is the same elliptic geometry (or a multiple cover thereof, e.g. a 3-sphere) and of an anti-de Sitter geometry it is a hyperbolic geometry. The problem comes in with that due to expansion, our universe is not homogeneous over all space and time. —Quondum 20:27, 5 May 2015 (UTC)

The Hyperbolic geometry#Geometry of the universe (including relativity) section is written on the mistaken assumption that Euclidean, hyperbolic and elliptic geometries are inherently 3-dimensional, which of course is rubbish. Sorry I don't have the expertise to correct it. Please could someone rewrite it to make sense? --Stfg (talk) 13:06, 26 May 2015 (UTC)

Not sure what you mean, but see the section #Geometry of the universe above. I don't understand why you think "The Hyperbolic geometry#Geometry of the universe (including relativity) section is written on the mistaken assumption that Euclidean, hyperbolic and elliptic geometries are inherently 3-dimensional". Space as we know it is three dimensional and time is something different from space. while they are both dimensions in space time they are not replaceable with each other, while the 3 dimensions of space are replaceable with each other. WillemienH (talk) 21:11, 26 May 2015 (UTC)
I've added a mention of time. Hopefully this will address the concern. The section references the three-dimensional cases of the geometry families plus time, but this is because of the dimensionality of our spacetime, not because of any assumption as suggested above. —Quondum 02:47, 27 May 2015 (UTC)
Thanks, that certainly helps. But the formulation
Describing our universe as three-dimensional (by thinking of it as Euclidean, elliptic or hyperbolic in addition to a separate time dimension) ...
still suggests that thinking of it as Euclidean, elliptic or hyperbolic in itself implies that it is three-dimensional. That is what misleads, since of course, all these geometries exist in arbararily many dimensions. It may seem pedantic, but many readers will have no idea that these geometries exist in higher dimensions, and I think many readers will go away with the notion that they are inherently 3-D. How about something like:
Describing our universe as a three-dimensional space (whether Euclidean, elliptic or hyperbolic) in addition to a separate time dimension ...
(Note that the parenthesis has moved). Sorry I'm in a bit of a rush today and am signing out now, but will be back tomorrow. Cheers, --Stfg (talk) 10:20, 27 May 2015 (UTC)
That would be an improvement. —Quondum 12:47, 27 May 2015 (UTC)
Thanks, I've done that and removed the tag. By the way, the section does need some references. --Stfg (talk) 10:30, 28 May 2015 (UTC)

## GA nomination and archiving

Hi

I nominated this page for good article (https://en.wikipedia.org/wiki/Wikipedia:Good_article_nominations ) As part of this i moved the complete talk page as now to /Archive 1 Later I will remove sections I find "out of date" "irrelevant"or "solved problems" and shorten discussions still old but still relevant. WillemienH (talk) 07:42, 26 June 2015 (UTC)

Removed some sections of this page as described above, they can still be found at /Archive 1 WillemienH (talk) 08:28, 26 June 2015 (UTC)

So where is the GA discussion? Here?
Also, the talk page content does not seem to be moved to the archive. YohanN7 (talk) 16:07, 17 July 2015 (UTC)

I am not sure about where the GA discussion should be hold, I guess we could keep it here. Or when you want to staer a reviwe of it on a dedicated review page (I am not sure how this works yet) About the archiving, I did remove the bits of this page that were (according to me) not any longer relevant. Bits of which i was not sure of the (present) relevancy I kept here. (but there are copies of them in the archive) WillemienH (talk) 20:11, 17 July 2015 (UTC)

## Lorentzian ≠ hyperbolic

This edit introduces the sentence "The geometry turns out to be hyperbolic because of Lorentz invariance." What is the intent? As it stands, it is false: the geometry of the four-dimensional Minkowski spacetime is not hyperbolic. Even a space-like flat section of it is a Euclidean 3-space. The hyperboloid model shows that a complete hyperbolic 3-space can be embedded isometrically in a Minkowski space, but that is clearly not the intent. The space of subluminal velocities (or equivalently, the space of time-like points at infinity) is a hyperbolic space, but this is as abstruse as saying that the space of directions in a Euclidean space is an elliptic geometry. So what is it that the source is saying? —Quondum 15:06, 28 June 2015 (UTC)

Undone now M∧Ŝc2ħεИτlk 16:46, 28 June 2015 (UTC)
From the edit summary: the sources talk about the line element and its hyperbolic nature. I do not have the source, but I guess this might be referring to the geometry of the space of lines. It would correspond to my point about the space of points at infinity. Since the article is about hyperbolic geometry, it may make sense to include a corrected form of this statement, along the lines of the space of velocities that I mentioned above – if that is what the source was saying. —Quondum 17:43, 28 June 2015 (UTC)
Okay, I've found a mention in the reference. In referring to a "line element", it is speaking of the (infinitesimal) difference of velocities in the space of velocities. I've re-added the statement as a stand-alone paragraph (it is quite distinct from the other geometries being referred to), reworded so that hopefully it will make sense to the lay reader. WillemienH, any comment would be welcome so that I can massage this statement for clarity and ease of understanding. —Quondum 14:36, 4 July 2015 (UTC)
Sorry for the very late reply, but my edit summary already summarized the response at least vaguely. The sources just talk about the invariance of the line element, that Lorentz boosts are hyperbolic rotations in spacetime, and mention that SR involves a non-Euclidean spacetime, without mentioning the geometry is hyperbolic. I have nothing more to say or intentions to clear up, sorry. M∧Ŝc2ħεИτlk 17:45, 4 July 2015 (UTC)
L&L is a bit more specific than that, now that I've managed to find the reference. To quote: "The required line element dl, is the relative velocity of two points with velocities v and v + dv" and "From the geometrical point of view this is the line element in three-dimensional Lobachevskii space—the space of constant negative curvature". There is also a formula for calculating dl2 in terms of v and dv only. To me, my edited re-insertion captures this clearly enough – they are saying that the distance measure makes the space of velocities into a three-dimensional Lobachevskian (i.e. hyperbolic) space, without any ambiguity. This is not to be confused with four-dimensional spacetime. It also fits exactly with the hyperboloid model, so it is a correct interpretation. The group of Lorentz transformations (including rotations) does act on this velocity space, just not in the same way as we are familiar with for normal spacetime. I was not asking for further help, but was merely asking whether someone feels that it needs further clarification, which I feel I could do if so. If not, it can stay as it is now. —Quondum 21:09, 4 July 2015 (UTC)
From your honored lay reader :) WillemienH (talk) 21:26, 4 July 2015 (UTC)
I do think that by now I knew something about hyperbolic geometry, but maybe you are talking about general relativity or space time, in which case you are correct, I try to stay away of it (I interest is in Geometry, not in Physics :), I think it all boils down to a confusion between Minkowski geometry and hyperbolic geometry, Minkowski geometry is an affine geometry, which hyperbolic geometry is not, And I am wondering is space time isotropic? (I guess not)
I stay away from editing the sections on "Geometry of the universe (including relativity)" and "The hyperboloid model" just because I am not familiar with them.
(I only started the section "Geometry of the universe (including relativity)" to save the "Geometry of the universe (pre dating relativity)" from edit wars, confusing the two.
I do think the hyperboloid model is quite useless for hyperbolic plane geometry as it is not embedded in euclidean space. (it is embedded in Minkowski space , but what does that mean, is Euclidean space not an Minskowski space ? , see here my knowledge is really in deficient) Lorentsian or not? I don't know, i don't even know what it means , so back to you , the not-so-lay person. :) WillemienH (talk) 21:26, 4 July 2015 (UTC)
I'm only a lay person myself, just with an interest in physics and mathematical structure, though I only discovered abstract geometry very recently, in particular Klein's approach. Physics gives only one application for geometry; one can tackle this entirely from a geometric perspective.
A four-dimensional Euclidean space can be projectively completed; the additional points (the points at infinity) form a three-dimensional elliptic space; I mention this as a framework that I assume you are familiar with. In a geometric sense, four-dimensional Minkowski space is not isotropic (in the linked sense; there is another sense of the word which means the opposite), since there is a light cone which separates timelike directions from spacelike directions. Similarly to the Euclidean case, the points at infinity form a geometry of is own, in this case two geometries: the one (timelike) set of points at infinity form a three-dimensional hyperbolic space, and the other (spacelike) set forms a three-dimensional de Sitter space. The crucial point as that the points at infinity are a different set (or space) from the points of the affine geometry in both the Euclidean and Minkowski cases. A Euclidean space is not a Minkowski space, even though both are affine spaces of the same dimension. A spacelike section of Minkowsi space is a three-dimensional Euclidean space, so in that sense Minkowski space contains Euclidean subspaces. It would be nice if somehow we can turn this discussion into text in the article to make it simple and understandable. Perhaps we should refer to the points at infinity rather than the space of velocities (these being equivalent)? —Quondum 04:51, 5 July 2015 (UTC)

A so we are both non cognanti in this. I am wondering about the whole relation between Minkowski geometry and Minkowski space-time and I don't think any of them gets any good of mixing the two up.

As you explain it now doesn't make any sense either (sorry I am being very critical here) If the hyperboloid model is embedded in an geometry that is not Euclidean, then the relations between the models loses its meaning. (because the other models are embedded in Euclidean geometry) Maybe I just don't understand minkowski geometry I am now reading a bit on it. but it looks like it has nothing to do with what is written at Minkowski space (if only lets start with it as only a geometry start with the 2 dimensional case, its relations with affine geometry and euclidean geometry. forgetting all about special relativity. If you like I can start with a stub idea. i found some (not a lot) links for a start of an articcle WillemienH (talk) 15:41, 5 July 2015 (UTC)

Minkowski spacetime and Minkowski geometry are the same thing. The hyperboloid model is isometrically embedded in Minkowski space. The Poincaré and Klein models are non-isometrically embedded in Euclidean space, which is to say the "euclidean" part of it is irrelevant. So in a sense, the hyperboloid model is more faithful than the Poincaré and Klein models are. My description should not be at odds with Minkowski space. —Quondum 17:50, 5 July 2015 (UTC)

I disagree Minkowski spacetime and Minkowski geometry are definitly not the same thing. (if anything Minkowski spacetime has 4 dimensions while Minkowski geometry also exists in 2 dimensions, found a nice link : [2] see nothing to do with spacetime at all. I am intrigued what do you mean by "The hyperboloid model is isometrically embedded in Minkowski space" ? (Do you mean the length of the curve is a measure of the distance between the points? or something else) WillemienH (talk) 22:12, 5 July 2015 (UTC)

Reviewing Cayley-Klein metric may assist in resolving geometric differences.Rgdboer (talk) 22:34, 5 July 2015 (UTC)
Rgdboer, sorry, that does not help me much, even though the content relates.
WillemienH, are you saying that you want to reserve "Minkowski space" to mean specifically the four-dimensional case, and "Minkowski geometry" to mean any number of dimensions)? (Or only two dimensions?) This is a rather trivial point of terminology and is not of concern to me. Yes, I mean the length of any curve on the hyperboloid is the same as measured in the hyperbolic space and as measured in the embedding space, using the Minkowski metric. The link you gave is quite cute, but I haven't read through all of it. But the gryphon is clearly talking about the same thing as I am. —Quondum 23:37, 5 July 2015 (UTC)

Hi Quondum, I moved (or restarted ) the discussion about Minkowski geometry at Talk:Minkowski space#Splitting off Minkowski geometry I think that is the better place to discuss it. (hope you can agree that there is the better place) I am not sure about your "Yes, I mean the length of any curve on the hyperboloid is the same as measured in the hyperbolic space and as measured in the embedding space, using the Minkowski metric." That's all to advanced for me. I noticed that something similar is written at the hyperboloid model but that did not mean anything to me either. (while I think I do understand distances in the other models, a bit) WillemienH (talk) 09:43, 7 July 2015 (UTC)

Actually, I need to check that statement on isometric embedding, though I have no reason to doubt it. I'll have to spend a little time doing the math from first principles; not complicated, but I'm rusty. It would help if my statement is checked and made explicit at Hyperboloid model. I'll leave some time for replies at the Minkowski space talk page; yes, that is a better space for such a discussion, though obviously you already know my feeling on the subject . —Quondum 13:01, 7 July 2015 (UTC)

## GA Review

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

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)

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)

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 ${\displaystyle u(z,w)={\frac {|z-w|^{2}}{4\Im {z}\Im {w}}}}$ then ${\displaystyle d(z,w)=arccosh(1+2u(z,w))}$. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)
• 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)
• Don't forget to multiply by radius of negative curvature squared JFB80 (talk) 20:56, 25 January 2016 (UTC)
• 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)
• 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)
tiling with right angled hexagons in the poincare disk model
• 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)
• 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)
• 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)
• 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)
• 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)
• The connections to relativity could be clarified better, in both places. --Sammy1339 (talk) 21:46, 13 August 2015 (UTC)
• 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).

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)

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)

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)

I agree with the addition from User:JRSpriggs about coordinate distance [3] 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)

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
${\displaystyle x=C}$
or
${\displaystyle \tanh y=A\cosh x+B\sinh x\quad {\text{ when }}\quad A^{2}<1+B^{2}}$
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)
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)

## Justifying the classification of isometries

In a hidden comment in the text of section Hyperbolic geometry#Isometries of the hyperbolic plane, WillemienH (talk · contribs) questions whether the classification includes all isometries which can be achieved with three reflections. I will explain the derivation of our classification.

First, I take it as obvious that the isometries of S1 (a one-dimension "sphere", i.e. a circle) are:

• identity (zero reflections, zero degrees of freedom),
• reflection (one reflection, one degree of freedom), and
• rotation (any two reflections separated by half the angle of the turn, one degree of freedom).

A third reflection just cancels one of the reflections already being used, changing the rotation back to a reflection. Therefor, the isometries of the hyperbolic plane (or of the Euclidean plane or of the 2-sphere) which hold a given point fixed will have the same classification.

Now consider an arbitrary isometry U of the hyperbolic plane. Choose any point p. If U(p)=p, then we are done because the above classification is a sub-classification of the one given in this article. If U(p)≠p, then draw a line segment from p to U(p). Find its midpoint and construct the perpendicular bisector. Let R be the reflection through that perpendicular bisector. R(U(p))=p by the construction of R. Since R is a reflection, the composition of R with itself RR=I is the identity. Thus U=URR. Let us now consider UR, see that UR(U(p))=U(p). That is, UR holds U(p) fixed. So we can apply the above classification to UR.

If UR is the identity, then U=URR=IR=R which is just a reflection through a line.

If UR is a reflection Q, then U=QR. Consider the two lines through which the reflections reflect. They cannot be the same because the line for R does not pass through U(p), but the line for Q does pass through U(p).

If the lines for Q and R intersect at the midpoint (between p and U(p)), then they are perpendicular and U is the inversion through that midpoint.

If the lines intersect elsewhere, then U is a rotation around that intersection.

If the lines do not intersect but are asymptotically parallel, then U is a "rotation" around the ideal point to which the lines tend.

If the lines are ultraparallel, then construct a third line perpendicular to both of them. Then U will be a translation along that third line.

If UR is a rotation around U(p), then we can choose two lines through U(p) separated by half the turn angle. Let the first of them be the line (of reflection S) which passes through p (and the midpoint). Then UR=TS, so U=TSR. Then SR will be the inversion through the midpoint. Drop a perpendicular from the midpoint onto the line of T. Let the lines of V and W be perpendicular intersecting at the midpoint (so that SR=WV) with the line of V being the perpendicular dropped onto the line of T. Then U=TWV and TW  is a translation along the line of V since the lines of T and W are both perpendicular to the line of V.

Is that clear? JRSpriggs (talk) 05:09, 18 July 2016 (UTC)

Sorry your answer is not very clear but also that was not my question (sorry I was not very clear either).
My question is more specific "Is every combination of 3 reflections a single reflection or a glide reflection, a glide reflection described as a combined reflection through a line and translation along the same line?".
Especially how can you prove that 3 reflections that don't combine to a single reflection (because the reflection lines are part of the same pencil) always combine to a glide reflection, a combined reflection through a line and translation along the same line? WillemienH (talk) 22:08, 18 July 2016 (UTC)
Yes, every composition of three reflections in the hyperbolic plane is equivalent to either a single reflection or a glide reflection.
I already proved it above, especially in the last paragraph. Just take U to be your composite of three reflections.
I thought about trying to give another kind of argument for you, but there are just too many cases to consider. You really need to use p and R to make the simpler argument work. JRSpriggs (talk) 05:21, 19 July 2016 (UTC)
Maybe it would help if I give an overview or executive summary of the relevant part of my proof. To your three reflections, I add a fourth carefully selected reflection. This reduces them to just a rotation, i.e. two reflections. To cancel out the change that I made when I added the fourth, I add it again (since it is its own inverse). Now we have a reflection and a rotation. I break the rotation into a reflection perpendicular to the reflection I added and another reflection. The two perpendicular reflections form an inversion. So we now have that your three reflections are equivalent to an inversion and a reflection. The inversion can be broken into a reflection perpendicular to the reflection and another reflection. So we now have two reflections, both perpendicular to a third reflection. The two form a translation along the line of the third, i.e. it adds up to a glide reflection. OK? JRSpriggs (talk) 08:46, 19 July 2016 (UTC)

Lemma: If A and B are reflections and p is any point, then there are reflections C, D and E such that p lies on the line of D, and AB=CD=DE.

Any two reflections form a rotation, horolation, or translation. If a rotation, let the line of D be the line connecting p to the center of the rotation. If a horolation, let the line of D connect p to the ideal point shared by A and B. If a translation, then let the line of D be the line one gets by dropping a perpendicular from p to the line along which A and B are translating. JRSpriggs (talk) 17:23, 20 July 2016 (UTC)

Suppose we have three reflections which I will call 1, 2, 3 (applied in that order). Pick any point q on the line of 3, but not on the line of 2. Using the lemma, shift 1 and 2 to 1' and 2' so that 2' goes thru q. Then 2' and 3 form a rotation about q (or the identity which reduces this to just a single reflection 1'). Drop a perpendicular from q onto the line of 1'. Call the foot of that perpendicular m. Now shift 2' and 3 to 2' ' and 3' so that 2' ' goes thru m. Then 1' and 2' ' meet perpendicularly at m, so they form an inversion thru m. Drop a perpendicular from m onto the line of 3', and call its foot n. Shift 1' and 2' ' to 1' ' and 2' ' ' so that 1' ' goes thru n. Then 1' ' and 2' ' ' still meet perpendicularly at m, but now 1' ' and 3' also meet perpendicularly at n. Thus 3∘2∘1=3∘2'∘1'=3'∘2' '∘1'=3'∘2' ' '∘1' ' is the composition of a reflection 1' ' and a translation 3'∘2' ' ' along the line of 1' ', i.e. they form a glide reflection. Do you understand now? JRSpriggs (talk) 13:33, 28 July 2016 (UTC)
Since the objective is to prove that the three reflections are equivalent to either one reflection or a glide reflection, I will assume that they are not equivalent to a single reflection. Thus 1≠2, because otherwise 3∘2∘1=3. And 2≠3, because otherwise 3∘2∘1=1. There is a point q on 3 which is not on 2, because otherwise 3=2. q is not the common element of 1 and 2, because otherwise it would be on 2. 2' is not 3, because otherwise 3∘2∘1=3∘2'∘1'=1'. m is not q, because otherwise 1' and 2' would share both the common element of 1 and 2 and the distinct point m=q so that 1'=2' and thus 1=2. n is not m, because otherwise 2' '=3' so 2'=3. I hope that fills in the gaps in the proof by eliminating any special cases which might cause problems. JRSpriggs (talk) 05:09, 29 July 2016 (UTC)

I copied the section Hyperbolic geometry#Isometries of the hyperbolic plane to Hyperbolic motion#Motions on the hyperbolic plane that page needed a more general (model independent) section and this section was just that. I guess in the long run the two copies will differ but thart is no worry to me. WillemienH (talk) 23:24, 8 September 2016 (UTC)

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