Talk:Poincaré disk model

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Previous discussion[edit]

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[edit]

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)

Isometries and Mobius transformations[edit]

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)

Metric[edit]

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)