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 $R$).

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.


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.

  • 1
    $\begingroup$ Thank you very much for your good explanation! I got it! I'm honoured to see that I got an answer from the creator of this awesome game! Keep up the good work! :) $\endgroup$ – ndrizza Jul 1 at 15:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.