# Talk:Poincaré metric

## Picture of J-invariant

What's up with the random picture of the J-invariant? The connection to this page is pretty opaque, although it's a beautiful picture. --Dylan Thurston (talk) 23:44, 4 June 2009 (UTC)

One might also point out that the upper half plane is effectively a disk of infinite diameter.

## Merger

There's a lot more information on the disk model over at Poincaré disk model. These two pages seem similar enough that I don't see a great need for two separate articles. --Dantheox 22:20, 30 April 2006 (UTC)

I'm against the merger, for multiple reasons: 1) Its OK to duplicate some information across multiple articles. 2) Both articles are already rather long; merging them would make them unmanageably long. 3) Neither article has exhausted what one can say about either topic: much more can be said about the disk, and much more can be said about the metric. 4) The Poincare disk model is about the N-dimensional hyperbolic model. This article (Poincare metric) limits itself to two dimensions only. Although we could generalize this article to N dimensions, there's actually a big difference between two and more than two dimensions: There's rigidity in three and higher dimensions, whereas in two, there is a very rich set of surfaces. One also looses some of the natural connections to number theory and modular forms and complex analysis. Etc. 5) combining them would lead to topic confusion, is the resulting combo about the disk or the metric? Suppose one wanted to talk about the metric on the half-plane? Does that mean we should merge with the half-plane article too? Before long, we'll end up merging a dozen articles because they all contain inter-related info. linas 01:59, 1 May 2006 (UTC)
That's fine.. perhaps a "Main article on this topic" link in the disk section of this article would be a more appropriate course of action. --Dantheox 04:52, 1 May 2006 (UTC)
I think the merger tags should be removed from both articles, as this idea isn't going anywhere. As for a main article link, it isn't clear that the other article is the main article on the metric; it's an article on the model, and mentions the metric very briefly. I'll add a link to the "see also" section. Gene Ward Smith 06:15, 1 May 2006 (UTC)

## Order in cross-ratio

Currently the article defines cross-ratio as ${\displaystyle (z_{1},z_{2};z_{3},z_{4})={\frac {(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{2}-z_{3})(z_{4}-z_{1})}}}$ which is inconsitent with the order defined in cross-ratio. I'll tag this inconsistency. I'm not sure whether the formula after that refers to the common definition of cross-ratio or actually relies on this non-standard definition. It might need adjustment as well. Someone should verify this. -- Martin von Gagern (talk) 15:31, 11 July 2011 (UTC)

## Missing factor of 2

I inserted a factor of 2 in the formula for ${\displaystyle \rho (z_{1},z_{2})}$. With the old formula we would have ${\displaystyle \rho (x,0)=x+O(x^{2})}$ for small positive ${\displaystyle x}$, which is clearly inconsistent with the formula for ${\displaystyle ds^{2}}$. The formula for ${\displaystyle ds^{2}}$ is the right one if we want to have curvature ${\displaystyle -1}$. Neil Strickland (talk) 13:07, 8 January 2015 (UTC)

## Cayley-Klein metric

Main article: Cayley-Klein metric

Though Poincare's name is attached to two models of the hyperbolic plane, the metric in these planes is named after Arthur Cayley and Felix Klein. Appropriate use of references may preserve this article for some applications in Riemann surface theory, but the reference to the hyperbolic plane models is incorrect. Currently there are no in-line references. Inaccurate statements are subject to change. — Rgdboer (talk) 21:30, 10 January 2016 (UTC)

My search for "Poincare metric" led to an article careful in use of terminology: H.S. Bear (1991) "Part metric and hyperbolic metric", American Mathematical Monthly 98: 109–123. In this case "metric of the Poincare model" does not translate to "Poincare metric". — Rgdboer (talk) 01:25, 11 January 2016 (UTC)

The source of this article is M%C3%A9trique_de_Poincar%C3%A9. One of our standards is WP:SET (search engine test) which yielded five times more for Poincare than Cayley-Klein, but then SET is known to be unreliable in cases like this. — Rgdboer (talk) 01:09, 13 January 2016 (UTC)

## Error: Poincare metric versus Cayley-Klein metric

${\displaystyle \rho (z_{1},z_{2})=2\tanh ^{-1}{\frac {|z_{1}-z_{2}|}{|z_{1}-{\overline {z_{2}}}|}}}$

is not equal to

${\displaystyle \rho (z_{1},z_{2})=\log(z_{1},z_{2};z_{1}^{\times },z_{2}^{\times }).}$

The former is the Poincare metric. The latter is the Cayley-Klein metric. Mosher (talk) 14:33, 29 February 2016 (UTC)

Cayley-Klein is a generic term for metrics from projective geometry. What source are you referring to with "Poincare metric" ? — Rgdboer (talk) 21:24, 29 February 2016 (UTC)
I find the whole article difficult to understand I think it could do with some rewriting or maybe even splitting of some of the information to a seperate articles:
${\displaystyle d(z_{1},z_{2})=\log(z_{1},z_{2};z_{1}^{\times },z_{2}^{\times }).}$ is a function for the distance on both the poincare disk model and the poincare half-plane model (depends a bit on the meaning of ${\displaystyle z_{1}^{\times },z_{2}^{\times }}$ )
${\displaystyle \rho (z_{1},z_{2})=2\tanh ^{-1}{\frac {|z_{1}-z_{2}|}{|z_{1}-{\overline {z_{2}}}|}}}$ looks a bit on the distance function of the centre of the poincare disk model to another point in the model, in general does not work for two points in the model.
That is all I can say about it. a description what the difference is between the meanings and uses of the Poincare metric and the Cayley-Klein metric would be welcome (in addition to the formula's, are they really two different metrics for the same model of hyperbolic geometry? and why do we need two?)
That's my 2 cents hope somebody more knowledgeable than me can act on them. WillemienH (talk) 10:06, 1 March 2016 (UTC)
The article refers to q-analogues where reference is made to umbral calculus, where there is no mention of q-analogues. The material elludes me. — Rgdboer (talk) 23:22, 2 March 2016 (UTC)
Maybe interesting here the question has been asked before and answered that they are similar: http://math.stackexchange.com/q/160338/88985 WillemienH (talk) 10:10, 4 March 2016 (UTC)