Talk:Hyperboloid model

wibni

I'd like to see some explicit examples of isometry matrices. Given the coordinates of a point, what is the matrix for rotation about that point? Given two points, what is the matrix for translation between them, or reflection in the line through them? —Tamfang 19:16, 8 April 2007 (UTC)

I added the matrix for rotations about a point in this edit. — Preceding unsigned comment added by TheKing44 (talk • contribs) 03:04, 6 September 2020 (UTC)

Okay, I added the matrix for the translation from one point to another in this edit. TheKing44 (talk) 13:36, 6 September 2020 (UTC)

history

The references given would appear to justify the statement made: "Being a commonplace model by the twentieth century ..etc.". However Jansen, who paper was published in 1909, the year of Minkowski's death, said that the reason for writing his paper on the hyperboloid representation was that he could nowhere find anything about it in the literature. It may be added that the representation was little known until recently. It was not described for example in the much quoted 1982 survey of hyperbolic geometry by Milnor in Bull. Amer. Math. Soc.JFB80 (talk) 16:58, 11 November 2010 (UTC)

Today I added Vladimir Varicak as an early proponent. As for Jansen's statement, "he could nowhere find anything" expresses his experience. It used to be difficult to survey literature as libraries had limited holdings; now we must partition our reading over many choices. Why Milnor neglected it doesn't matter. There are other old references still to be posted.Rgdboer (talk) 00:21, 16 December 2010 (UTC)

Li,Hestenes,Rockwood (A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces) give W. Killing. Ueber zwei Raumformen mit constanter positiver Krummung. J.Reine Angew. Math., 86:72–83, 1878 and say " Killing described a hyperboloid model of hyperbolic geometry by constructing the stereographic projection of Beltrami’s disc model onto the hyperbolic space. This hyperboloid model was generalized to n-dimensions by Poincare." Selfstudier (talk) 15:41, 5 March 2012 (UTC)

It sounds doubtful to me having actually taken the trouble to look at original papers of Killing and others. It is strange how people can be so wise without checking for themselves.JFB80 (talk) 18:49, 5 March 2012 (UTC)

You could well be right, I can't read German and don't have the paper anyway. What is Wikipedia policy? Can we just ignore sources that we think might be wrong? How would we decide which are right and which wrong?Selfstudier (talk) 08:40, 6 March 2012 (UTC)

I did notice was that there did seem to be quite a bit of confusion in the different sources (I found some others), although the Poincare 1880/1 seems quite solid...Selfstudier (talk) 08:40, 6 March 2012 (UTC)

I got this from an Arxiv paper [1]:

The first researcher who mentioned Karl Weierstrass in connection with the hyperboloidal model of the Bolyai-Lobachevsky plane was Wilhelm Killing. In a paper published in 1880, [40], he used what he called Weierstrass’s coordinates to describe the “exterior hyperbolic plane” as an “ideal region” of the Bolyai-Lobachevsky plane. In 1885, he added that Weierstrass had introduced these coordinates, in combination with “numerous applications,” during a seminar held in 1872, [42], pp 258-259. We found no evidence of any written account of the hyperboloidal model for the Bolyai-Lobachevsky plane prior to the one Killing gave in a paragraph of [42],p. 260. His remarks might have inspired Richard Faber to name this model after Weierstrass and to dedicate a chapter to it in [29], pp. 247-278.

And this from Worlds out of Nothing:

What follows is a modernised summary of what Killing went on to describe in his Die nicht-Euklidischen Raumformen (The non-Euclidean space forms), concentrating on the two-dimensional case – Killing naturally discussed the n-dimensional situation. The Weierstrass–Killing approach was to start with a surface with equation k^2z^2 +x^2 +y^2 = k^2 , where k^2 is taken as the curvature of the surface. So when k^2 = 0 the surface is a plane, when k^2 = +1 the surface is a sphere, and when k^2 = − 1 the surface is a hyperboloid of two sheets. The basic idea was to mimic trigonometry on a sphere. Selfstudier (talk) 14:12, 6 March 2012 (UTC)

Seems like everybody would like to have this model named after them:-)....Selfstudier (talk) 14:14, 6 March 2012 (UTC)

I would like to comment on your previous remark on the Weierstrass-Killing approach. You say 'when k^2 = − 1 the surface is a hyperboloid of two sheets'. That is true but the geometry of this surface is not hyperbolic (Lobachevskian) because the Gaussian curvature is zero as it is a ruled surface. This looks like a mistake in the thinking of these early papers. JFB80 (talk) 04:28, 7 May 2015 (UTC)

It doen't make any sense "The Weierstrass–Killing approach was to start with a surface with equation k^2z^2 +x^2 +y^2 = k^2 , where k^2 is taken as the curvature of the surface." No, the gaussian curvature of such a surface is just not k^2, but some complicated formula. Also "a hyperboloid of two sheets ... is a ruled surface" No its not! An hyperboloid of ***one*** sheet is a ruled surface (and the curvature of an hyperboloid of two sheets is not constant and nowhere negative. And so i guess a couple of more disputable claims (ruled surfaces have no curvature ? ) can posters here be a bit geometricly inclined??? WillemienH (talk) 06:30, 7 May 2015 (UTC)

True - it doesnt make sense. It was just a slip on my part. But, whatever the meaning of k, the surface is a hyperboloid. Now a 2 sheeted hyperboloid has positive curvature and a 1 sheeted hyperboloid, which looks as though it has negative curvature, is not Lobachevskian as you can find out either by doing a long calculation for curvature or by observing that it is a ruled surface so has zero curvature.JFB80 (talk) 15:51, 7 May 2015 (UTC)

Where do you get that a ruled surface has zero intrinsic curvature? It is a straightforward exercise to construct examples of ruled surfaces with negative curvature in Euclidean space. A hyperboloid of one surface and a hyperboloid of two surfaces both have nonzero curvature, respectively positive and negative by the inferred metric from the space in which they are embedded, provided that that space is Euclidean. However, the argument is spurious in this context, as the embedding space does not have a Euclidean metric in the hyperboloid model; the intrinsic curvature of a hyperbolic space is negative, not positive. The appropriate metric of the embedding space is probably the Minkowski metric for this model. —Quondum 16:54, 7 May 2015 (UTC)

The point about zero Gaussian curvature is really irrelevant to what I was trying to say. The article traces the model back to Weierstrass (1870) and Killing (1880, 1875) who wrote well before Minkowski (1908) invented his space-time with the pseudo-metric which gives the 'hyperboloid model'. So what were Weierstrass and Killing saying? It was apparently that the hyperboloid in Euclidean space gives a Lobachevsky space which it does not. 94.66.68.196 (talk) 18:59, 7 May 2015 (UTC)

Agreed that Gaussian curvature is not relevant, though even that is nonzero (it is the product of the principle curvatures, which are not along the straight lines). I rather doubt whether people like Weierstrass and Killing would have made the mistake of thinking of the model embedded in a space with a Euclidean metric. I see nothing to suggest that they did. I'm not too sure what you're getting at. —Quondum 20:26, 7 May 2015 (UTC)

You say you rather doubt. But of course they were embedding in Euclidean space in those days. What is your explanation? That is what I am asking. To quote from a previous comment by Selfsudier,"The Weierstrass–Killing approach was to start with a surface with equation k^2z^2 +x^2 +y^2 = k^2". Is not this a hyperboloid when k^2 is negative, which case they considered? How could they come to the conclusion that it represents a Lobachevsky space? Please dont dodge this obvious question, please at least admit it is something strange which needs to be understood and not just passed by.JFB80 (talk) 01:37, 8 May 2015 (UTC)

They were considering it as embedded in a vector space or an affine space, as suggested by the coordinates x, y, z. This does not imply a Euclidean metric. Unfortunately, people sometimes mean an affine space when they say "Euclidean space". Embedding the hyperboloid in an affine space with a Minkowski metric gives a hyperbolic geometry (a Lobachevsky space). I have been careful in what I've said above to use the term "Euclidean metric" so that there would be no confusion. If you understand the hyperboloid model, I can't see how you would be making wild assertions such as you are making. It is a real model. It works. And you'd understand how a hyperboloid of two sheets has a (non-constant) positive curvature if embedded in an vector space with a positive-definite (Euclidean) quadratic form, but a constant negative curvature if embedded in an vector space with a Minkowski quadratic form, as given in the article. If we get the same conclusion as they did, it would be arrogant to assume that they did not know enough to reach the same correct conclusion using valid reasoning. —Quondum 02:06, 8 May 2015 (UTC)

After reading remarks of Reynolds in Bull AMS 1993 I am seeing the history more clearly. Reynolds said that Weierstrass 1880 did not discuss hyperbolic geometry but only introduced the Weierstrass coordinates (which give coordinates for a sphere of imaginary radius). It is this sphere of imaginary radius idea that runs through and unifies the history. It also explains how Minkowski first came to the idea because initially he used an imaginary time coordinate ict and then his imaginary version of space-time can be given a similar interpretation. Then with his later 'Space and Time' affine geometry version, he introduced the pseudo-metric. It has become the practice to use the historical background to justify the use of this representation avoiding direct derivation that the 2 sheeted hyperbola has hyperbolic geometry for the Minkowski pseudo-metric. I have not seen this demonstrated anywhere in the more recent literature even in Reynold's article though he describes the history very well, better than Wikipedia. Thank you for your comments.JFB80 (talk) 18:40, 8 May 2015 (UTC)

If you have taken the trouble to check the detail (and your summary here sounds cohesive), then you might want to adjust what is in the article to how the Reynolds article it? —Quondum 19:18, 8 May 2015 (UTC)

Was Minkowski space named after him because he invented it, or merely because he was the first to realize that special relativity implied that it is the space in which we live? I suspect that Minkowski space was known as a mathematical abstraction before Minkowski's realization, just not called by that name. JRSpriggs (talk) 10:58, 9 May 2015 (UTC)

In answer to Quondum:I may adjust the article in due course but would like to think about it further. To JRSpriggs' question I think the answer is that it was after Minkowski's 1909 paper 'Space and Time' that relativity people e.g. Sommerfeld, started to talk about Minkowski space-time. But as I said above Minkowski had previously used the same idea in its complex form; the surface x^2+y^2+z^2+(ict)^2 = negative constant occurring in the theory corresponding exactly with the form quoted by Killing from Weierstrass. Killing's 1880 paper can be found on https://gdz.sub.uni-goettingen.de/ (search for Killing then order papers by name)I think its far better to check with the original source than to quote second- and third-hand accounts, which seem to be getting things quite wrong. Even if you cant read German the equations tell the story.JFB80 (talk) 16:50, 10 May 2015 (UTC)

Add a picture

Someone find a pic of the 3D hyperboloid space. The topology of the klien and poincare models are intuitively obvious from just looking at the images of 3D tesselations of those spaces. The pictures are also very beautiful. Well, to a geek. —Preceding unsigned comment added by 75.108.188.137 (talk) 06:55, 6 January 2011 (UTC)

I don't understand exactly what is requested/suggested here. —Tamfang (talk) 18:28, 6 January 2011 (UTC)

I think he wants something like Klein or Poincaré. But since the hyperboloid model is a three dimensional surface in a four dimensional space, it is not clear to me how we could make a picture of it even if we imposed some regular structure on it to provide content. JRSpriggs (talk) 22:32, 6 January 2011 (UTC)

You can do a 2/3D version with projection to eg Poincare disc

Add caption here

Selfstudier (talk) 13:51, 18 February 2012 (UTC)

The image was posted because its got the basic idea. To be correct the hyperboloid should be tangent to the disk, and the asymptotic cone should include the boundary of the disk. If the origin is the center of projection, then the disk rests at level z = 1. Hyperboloid: x^2 + y^2 − z^2 = 1. Rgdboer (talk) 03:01, 19 February 2012 (UTC)

Your equation above is for a hyperboloid of 1 sheet, not two; in the set up I posted the centre of projection is at z = -1, a disk tangent at 1 would be the Klein disk and the hemisphere model between the two disks all related by projection.Selfstudier (talk) 10:24, 19 February 2012 (UTC)

Something like the description hereSelfstudier (talk) 10:37, 19 February 2012 (UTC)

Tangency is not necessary; changing the disk's height changes only the scale of the projection. —Tamfang (talk) 22:20, 5 March 2012 (UTC)

Using the same set up as in the picture ie with 1 and centre of projection at 0, the Klein disk is tangent. Selfstudier (talk) 08:27, 6 March 2012 (UTC)

Selfstudier (talk) 11:42, 7 March 2012 (UTC)

Selfstudier, you are right about my comments. Thank you for your informative illustrations. Including the hemisphere model is a good idea, not yet developed for WP. Hyperboloid and hemisphere models complement each other in the set of models.Rgdboer (talk) 00:16, 15 March 2012 (UTC)

Redirect

I visit this page because I was redirected by clicking "Lorentz model" of dielectric permitivity. I am sure that "Lorentz model" I am looking for do nothing with this hyperboloid model. Would someone add a disambiguation to this page? —Preceding unsigned comment added by 128.100.75.116 (talk) 21:57, 24 March 2011 (UTC)

Dielectric permittivity doesn't mention Lorentz ... —Tamfang (talk) 23:25, 24 March 2011 (UTC)

However, Permittivity has a link to Lorentz model when then redirects here. This appears to be a mistake of some kind. JRSpriggs (talk) 13:59, 25 March 2011 (UTC)

The Permittivity article implies that the Lorentz model is a common generalization of the Debye model and the Drude model. But this is outside my area of knowledge. JRSpriggs (talk) 14:17, 25 March 2011 (UTC)

The redirect was put in by someone writing about metamaterials. See also the book Metamaterials: Physics and Engineering Explorations. The Lorentz model referred to has nothing to do with the hyperboloid model of hyperbolic geometry, so the redirect is incorrect, it should be deleted.Rgdboer (talk) 20:49, 25 March 2011 (UTC)

The only way I know of deleting an article (including a redirect) is to ask an administrator to do it. However, a better solution, if possible, would be to edit it to point to an article which is a more appropriate target. But I do not know the subject well enough to find such a new target. I left a message at the Physics Project which I hope will lead to it being fix by someone who does know. JRSpriggs (talk) 05:26, 26 March 2011 (UTC)

Drude model article mentions Lorentz-Drude model. Redirect has been changed today.Rgdboer (talk) 00:43, 15 March 2012 (UTC)

The form $\cosh A+\alpha \ \sinh A$

for the equation of points on the hyperboloid sheet mentioned in the article is reminiscent of a quaternion - a scalar combined with a (scaled) vector. Is there a connection worth making? —Preceding unsigned comment added by 78.146.15.138 (talk) 13:37, 16 May 2011 (UTC)

No. The only thing that would justify mentioning the quaternions is if their multiplication applied here, but it does not. JRSpriggs (talk) 19:21, 16 May 2011 (UTC)

The hyperbolic quaternion multiplication is used in Macfarlane's derivation of the hyperbolic law of cosines. Rather than mention quaternions or hyperbolic quaternions, the link to hyperbolic quaternions has been made through existing text and a piped link to hyperbolic quaternions as Algebra of Physics, the designation of the ring used by him. In this way readers can find some historical material without undue distraction.Rgdboer (talk) 02:38, 17 May 2011 (UTC)

Relation to Conformal Geometric Algebra

In Conformal geometric algebra the Minkowski space R^n,1 embedded into R^n+1,1 provides another convenient model for studying conformal transformations since the conformal group of n-dimensional Euclidean, spherical and hyperbolic spaces are all isometric to the group of isometries of hyperbolic (n+1)-space (as well as isometric to each other)Selfstudier (talk) 14:14, 7 March 2012 (UTC)

Alternative image

The following image was contributed by Zou170:

The hyperboloid model can be used to construct a Poincaré disk model. The red balls are arranged on a geodesic on the hyperboloid, the blue line is a geodesic on the Poincaré projection

The previous image was restored because it supports the Hyperboloid model with the Poincare disk model rather than the reverse. Learning hyperbolic geometry usually starts with Poincare half-plane and Poincare disk models, for instance in connection with Mobius transformations and inversive geometry. The reverse approach of the Zou710 image does not serve the general reader.Rgdboer (talk) 20:03, 18 October 2014 (UTC)

cosh^-1 or arcosh?

before we get an edit war: I changed the previous :$d(u,v)=\cosh ^{-1}(B(u,v)).$ to :$d(u,v)=\operatorname {arcosh} (B(u,v)).$ which was reversed by user:Arcfrk back to  :$d(u,v)=\cosh ^{-1}(B(u,v)).$

I suppose that with $=\cosh ^{-1}(x).$ is ment the Inverse hyperbolic function of cosh and that is generally described as $\operatorname {arcosh} (x)$ or was I mistaken here? I made a similar edit at Poincare disk model so plese et me know if i was wrong then i can learn from it . WillemienH (talk) 06:20, 27 July 2015 (UTC)

I already reverted the edit by the time I read this edit talk page comment (though I added a clarifying edit comment). arcosh is correct, is already used extensively in WP and in texts, and is unambiguous notation, as may be seen from articles such as Inverse hyperbolic function that I linked. As such, IMO it is to be preferred. A lack of familiarity on the part of some is not a reason to avoid it. —Quondum 13:27, 27 July 2015 (UTC)

Tensor

The following was removed as unreferenced:

The metric tensor for this space in the (x₁, x₂, ..., x_n) coordinate system (x₀ being included implicitly) is given by:

$g_{ab}=\delta _{ab}-{\frac {x_{a}x_{b}}{1+\sum _{k=1}^{n}x_{k}^{2}}}=\delta _{ab}-{\frac {x_{a}x_{b}}{x_{0}^{2}}}.$

The foundations of differential geometry have not been set for this comment. — Rgdboer (talk) 01:17, 9 November 2019 (UTC)

Geschwindigkeitsvectoren: Definition in Minkowski's paper apparently incorrect

The definition of velocity vector (Geschwindigkeitsvector) given in Minkowski's quoted 1907 paper appears to be incorrect (it is not Lorentz invariant) Minkowski redefined it in his 1908 paper 'Fundamental Equations ...' p.369 giving the definition now currently used (see e.g. Landau & Lifschitz equation (7.2)) JFB80 (talk) 16:03, 13 December 2019 (UTC)

Major edit adding matrices for rotations, translations, reflections, and hororotations

I made a major edit adding matrices for various transformations. I think they are useful, but I think we should consider some reformatting. I think some of the content above it is now redundant, but I didn't feel comfortable unilaterally removing it. Thoughts? TheKing44 (talk) 02:31, 6 September 2020 (UTC)

I noticed that the sections I added mostly correspond to subgroups. Maybe rephrasing it to be about subgroups of O⁺(1, n) would be good? TheKing44 (talk) 17:19, 6 September 2020 (UTC)

Minkowski space metric signature

(copying from user talk:jacobolus#For your interest)

https://arxiv.org/abs/2111.07353 An earlier discussion Selfstudier (talk) 14:33, 2 August 2022 (UTC)

Thanks Selfstudier. I agree, the choice of signature is ultimately arbitrary in the context of the hyperboloid model. I think it was fair for D0SBoots to change it here for clarity and my impression is that the -+++ version is a bit more common in mathematics (while +--- is more common in certain parts of physics). I tried to re-word D0SBoots’s addition; that +--- “does not work” is certainly not a fair or accurate claim, but it would be fair to say that it runs counter to conventions from some fields. I’m not sure how much it’s worth belaboring the point here. But I do wish Wikipedia had a better article about the squared distance (or "scalar square", "quadrance", etc.). There is discussion at Pseudo-Euclidean space but it seems worthy of its own topic. –jacobolus (t) 14:45, 2 August 2022 (UTC)