I'm trying to find a reference to the following statement.

Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar coordinates) by sending the point at hyperbolic distance $R$ from the origin at an angle $\theta$ to the positive $y$-axis to the point at Euclidean distance $R$ from the origin at an angle $\theta$ to the positive $y$-axis.

The rationale for why this should be $1$-Lipschitz is that the circumference of a circle of radius $R$ is $2\pi\sinh(R)$ in the hyperbolic plane and $2\pi R$ in the Euclidean plane, and $2\pi\sinh(R)\geq 2\pi R$ for all $R\geq 0$.

Unpacking the definitions of the metrics requires one to prove that \begin{equation} \cosh^{-1}\left(\cosh(r_1)\cosh(r_2)-\sinh(r_1)\sinh(r_2)\cos(\theta)\right) \geq \sqrt{r_1^2+r_2^2-2r_1r_2\cos(\theta)}. \end{equation} for all $r_1,r_2\geq 0$ and $\theta \in (-\pi,\pi]$, so I'm hoping there's a nicer way!

  • 1
    $\begingroup$ Your map is just the inverse of the exponential map $T_oH^2\to H^2$. Your map is 1-Lipschitz because of the hinge inequality. $\endgroup$ Commented Feb 27, 2023 at 12:24
  • $\begingroup$ Sorry @MoisheKohan I don't understand your comment. I see the map I'm interested in is the inverse of the exponentiation map as above, but the hinge inequality (at least the version I can find online) is only for changing angles in Euclidean spaces, why would it help here? $\endgroup$
    – DavidHume
    Commented Feb 28, 2023 at 21:26

1 Answer 1


I have been informed that the result I was after is a special case of Lemma I.1.13 of "Metric spaces of non-positive curvature" by Bridson-Haefliger.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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