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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!

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    $\begingroup$ 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

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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.

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