Talk:Poincaré disk model

Previous discussion
The Poincaré metric article seems to be more general, and includes the disk model in its discussion. This article has a lot more details on that specific model, but nothing that wouldn't fit into the larger setting. --Dantheox 22:26, 30 April 2006 (UTC)

The Poincaré metric article is by no means more general; in a lot of ways less general. It focuses on the hyperbolic plane, and on complex analysis. Merging the two would be difficult, and in general a bad idea, I think. Gene Ward Smith 06:04, 1 May 2006 (UTC)


 * I went ahead an removed the merge tags. It still seems like there's a significant amount of overlap between the two, however, and I'm sure that each could benefit from incorporating some of that material from the other. For example, the Poincaré metric page simply says that the geodesics are "circular arcs whose endpoints are orthogonal to the boundary of the disk." This article goes into lots more detail, with specific equations. --Dantheox 07:06, 1 May 2006 (UTC)

Should the Poincaré metric article also discuss the higher-dimensional case or what is the intention? Pierreback 21:12, 2 May 2006 (UTC)

Poincaré ball model
Sometimes the Poincaré model is called the Poincaré ball model or the conformal ball model. Perhaps the Poincaré ball model is a better name than Poincaré disk model? The name "ball model" clearly shows that the article also treats the higher-dimensional cases. Pierreback 21:11, 2 May 2006 (UTC)


 * against I just prefer disk model, there can be a section on higher-dimensional cases. but keep it as disk model. WillemienH (talk) 17:09, 29 August 2015 (UTC)

Isometries and Mobius transformations
Given Klein's geometry program, a discussion of the isometries being Mobius transformations would be nice. —Preceding unsigned comment added by 75.168.185.102 (talk) 22:08, 25 January 2009 (UTC)
 * The isometry group was identified as SU(1,1) today. — Rgdboer (talk) 22:12, 2 July 2019 (UTC)

Metric
I don't understand something here, may be someone can answer this:

If for n = 2 one uses the coordinates $$\vec{w} = \left(\begin{array}{c} x_1 \\ x_2 \end{array} \right) $$ with $$x_i = \frac{2 y_i}{1-y_1^2-y_2^2} $$ and calculates the metric $$ g_{ij} = \langle{\frac{\partial \vec{w}}{\partial y_i}}, {\frac{\partial \vec{w}}{\partial y_j}}\rangle$$ one gets

$$g_{11} = \frac{4 \left(y_1^4+\left(-1+y_2^2\right){}^2+2 y_1^2 \left(1+y_2^2\right)\right)}{\left(1-y_1^2-y_2^2\right){}^4} $$

$$g_{12} = g_{21} = \frac{16 y_1 y_2}{\left(1-y_1^2-y_2^2\right){}^4} $$

$$g_{22} = \frac{4 \left(y_1^4+2 y_1^2 \left(-1+y_2^2\right)+\left(1+y_2^2\right){}^2\right)}{\left(1-y_1^2-y_2^2\right){}^4} $$

which is not the metric mentionned in the article, that is:

$$g_{11} = g_{22} = \frac{4}{\left(1-y_1^2-y_2^2\right){}^2} $$

$$g_{12} = g_{21} = 0 $$

So what is wrong here? Ulrich Utiger 23 March 2012


 * What is right with it? What do your variables mean? Where did you get the equation for x in terms of y? What about t? JRSpriggs (talk) 06:30, 24 March 2012 (UTC)

I found the problem: the metric above is the metric of the regular surface of a hyperboloid, of which the general equation in three dimensions is

$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = -1 $$.

Setting a = b = c = 1 and x = x1, y = x2 we get the above metric with the coordinates of the hyperboloid (graph of the regular surface)

$$ \left(x_1,x_2,\sqrt{1+x_1^2+x_2^2}\right) $$

using the coordinate transformation $$x_i = \frac{2 y_i}{1-y_1^2-y_2^2} $$ (i = 1,2) and $$ t = x_3 = \frac{1+y_1^2+y_2^2}{1-y_1^2-y_2^2} $$. So these are not the coordinates but only a coordinate transformation.

The Poincaré metric on the other hand comes from the hyperbolic (or Minkowski) line element (or metric) $$ ds^2 = dx_1^2 + dx_2^2 - dx_3^2 $$. If we put the above coordinate transformation in it, we get the usual Poincaré metric. In fact, if one changes x3 into i*x3 one gets the coordinates $$\vec{w}$$ from the coordinate transformation.

I think that this article needs some clarification. It is too abstract and short for a real understanding. If only the specialists understand it, then it is not very useful. As a physicist, I would prefer that this work be done by a mathematician rather than by me ... Ulrich Utiger (talk) 10:32, 25 March 2012 (UTC)

Unreferrenced inversive geometry
The following was removed:

Isometric Transformations
The analog of a reflection about a line in hyperbolic space is a reflection about a geodesic, which can be represented in the model as a circle inversion about the circle that represents the geodesic. Rotations and translations can be represented as a combination of two reflections about different geodesics. In the case of rotations, the two geodesics intersect, while in the case of translations, they do not.

One result of this is that if hyperbolic space is translated such that the origin in the unit Poincaré disk is translated to $$\mathbf{v}$$, $$\mathbf{x}$$ is translated to


 * $$\frac{ ( 1 + 2 \mathbf{v} \cdot \mathbf{x} + \left| \mathbf{x} \right| ^2 ) \mathbf{v} + ( 1 - \left| \mathbf{v} \right| ^2 ) \mathbf{x}}{ 1 + 2 \mathbf{v} \cdot \mathbf{x} + \left| \mathbf{v} \right| ^2 \left| \mathbf{x} \right| ^2 } .$$

This also applies for higher dimensions.(end of removed text)

Algebraic expression of the isometry group by SU(1,1) has been provided. — Rgdboer (talk) 22:12, 2 July 2019 (UTC)

Distance Section Incorrect?
The "Distance" section says "Distances in this model are Cayley–Klein_metric." It also describes Klein disc model distances there. This is inconsistent with the metric described in the "Metric and curvature" section. I suspect that the "Distance" section was accidentally copied from another article and never updated for the model described in this article. — Preceding unsigned comment added by 146.115.176.205 (talk) 21:14, 3 January 2021 (UTC)

Incorrect edit?
Could someone have a look at this edit? Unless I messed up my implementation, the new formula is not correct.

Streetmathematician (talk) 10:22, 24 October 2021 (UTC)

There he describes a world, now known as the Poincaré disk
In the quotation that follows, he actually describes the 3D version of it. 2A01:CB0C:CD:D800:9587:C8EA:B22B:6D0A (talk) 10:49, 17 February 2023 (UTC)