$1$-Lipschitz map from hyperbolic to Euclidean plane

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 $$$$\cosh^{-1}\left(\cosh(r_1)\cosh(r_2)-\sinh(r_1)\sinh(r_2)\cos(\theta)\right) \geq \sqrt{r_1^2+r_2^2-2r_1r_2\cos(\theta)}.$$$$ for all $$r_1,r_2\geq 0$$ and $$\theta \in (-\pi,\pi]$$, so I'm hoping there's a nicer way!

• 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. Commented Feb 27, 2023 at 12:24
• 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? Commented Feb 28, 2023 at 21:26