Emulating distance on Poincaré disk for different curvatures

Question

The distance $d_K^H(x,y)$ between two points on the hyperboloid $H_K$ with curvature $K<0$ can be emulated on the distance $d_{-1}(x,y)$ of the hyperboloid $H_{-1}$ of curvature ($K=-1$) as follows: $$ d_K^H(x,y)=R\cdot d_{-1}^H(x/R,y/R) $$ where $R$ is the radius and is related to the curvature as follows: $R=\frac{1}{\sqrt{-K}}$.

Do you know about a simple formula to do a similar emulation with the distance $d_K^D(x,y)$ on the Poincaré disk $D_K$ of curvature $K$?

For $K=-1$ the distance on the Poincaré disk $D_{-1}$ is: $$ d_{-1}^P(x,y) = arccosh\left( 1+\frac{2||x-y||_2^2}{(1-||x||^2_2)(1-||y||_2^2)} \right) $$ So I'm looking for an expression of the form: $$ d_K^P(x,y)=\cdots d_1^P(\cdots x\cdots, \cdots y\cdots). $$ where the $(\cdots)$-parts are just replaced with some function or expression in terms of $K$ (or $R$).

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So far I've tried to project the points from the hyperboloid to the Poincaré disk. But it didn't turn out to be a nice expression.

Answer 1

The radius here seems to mean two things:

1. the radius of curvature, whose square root is inversely proportional to the Gaussian curvature,
2. the scale of the model (for example, the radius of the Poincaré disk, or a parameter of the hyperboloid, or the radius of a sphere represented in the $(x,y,z)$ coordinates).

If you have a distance formula $d(\overrightarrow{x})$ for a model of radius $R$, then you can define a distance formula on a model of radius $RC$ with $d'(\overrightarrow{x}) = d(\overrightarrow{x}/C)$. The new model is clearly isometric to the old model, so this does not change the Gaussian curvature.

On the other hand, if you define $d'(\overrightarrow{x}) = Cd(\overrightarrow{x})$, this multiplies the radius of curvature by $C$. Just like on a sphere: if you take the distance formulas for a sphere of radius 1 (in whatever coordinates), and multiply them by 10, you get the distance formulas for a manifold isometric to a sphere of radius 10, and thus with a smaller curvature, but still parametrized as if it was a sphere of radius 1.

So the $/R$ part in your hyperboloid formula controls the scale of the model, and the $R\cdot$ part controls the Gaussian curvature. If you want a metric on the Poincaré disk of radius 1 that makes the curvature different, just multiply your formula by a constant. If you also want to change the radius of the Poincaré disk for some reason, then you have to also adjust the coordinates.