There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $x\_1^2+x\_2^2-x\_3^2=-1$ embedded in a Minkowski space with metric $ds^2 = dx\_1^2+dx\_2^2-dx\_3^2$. If the hyperboloid is parameterized with the embedding


$$ 
x\_1(t,\phi)=\sinh t \cos\phi\\
x\_2(t,\phi)=\sinh t\sin\phi\\
x\_3(t,\phi)=\cosh t
$$


one finds the induced metric for the hyperbolic plane $ds^2=dt^2+\sinh^2t\,\,d\phi^2$.
Two further familiar coordinate systems include the projection onto the Poincare disk, given by the transformations


$$
y\_i = \frac{x\_i}{1+x\_3} \text{ for } i \in \{1,2\}
$$


The metric on the Poincare disk is given by $ds^2 = \frac{2}{1-\vec y^2}(dy\_1^2+dy\_2^2).$ Although it is called projection, it is actually invertible and therefore a regular diffeomorphism.


Finally, inversion on a circle with radius $r=\sqrt{2}$ centered at $(0,-1)$ projects the Poincare disk onto the upper half plane. The transformation is again a diffeomorphism


$$
z\_1 = 2\frac{y\_1}{y\_1^2+(y\_2+1)^2}\\
z\_2 = 2\frac{y\_2+1}{y\_1^2+(y\_2+1)^2}-1
$$


For the sake of completeness, the metric on the upper half plane is given by $ds^2 = \frac{1}{z\_2^2}(dz\_1^2+dz\_2^2).$


I am interested in a fourth representation of the hyperbolic plane. It is only an immersion, only a patch of the hyperbolic plane is covered. But on the other hand it is immersed into a regular Euclidean space with positive signature $ds^2 = dx\_1^2+dx\_2^2+dx\_3^2$.


The immersion is described by
$$
x\_1(T,\Phi) = \sqrt{1-{\rm e}^{2T}}-\text{arctanh}\sqrt{1-{\rm e}^{2T}}\\
x\_2(T,\Phi) = {\rm e}^T \cos\Phi\\
x\_3(T,\Phi) = {\rm e}^T \sin\Phi
$$


and the corresponding metric is given by $ds^2 = dT^2+{\rm e}^{2T} d\Phi^2$.


Does anyone know the coordinate transformation $T(t,\phi)$ and $\Phi(t,\phi)$, that directly transforms from the metric $ds^2 = dt^2+\sinh^2 t\,\,d\phi^2$ to the metric $ds^2 = dT^2+{\rm e}^{2T} d\Phi^2$? It doesn't look like an overwhelmingly difficult problem, but the resulting set of partial differential equations doesn't offer any clue for a possible solution:


$$
t\_{,T}^2+\sinh^2 t\,\, \phi\_{,T}^2 = 1\\
t\_{,T}t\_{,\Phi} +\sinh^2 t\,\, \phi\_{,T}\phi\_{,\Phi}=0\\
t\_{,\Phi}^2+\sinh^2 t\,\, \phi\_{,\Phi}^2 = {\rm e}^{2T}
$$


Is it possible to perform a Wick rotation and get from the first embedding to the immersion?
It also has some resemblance to the different slices of de-Sitter space-time for open and flat three-dimensional sections, but it is not a much help either to me, unfortunately.



Coordinate transformation for the hyperbolic plane to the pseudo sphere
In the *pseudosphere* representation you wrote down, $T\in (-\infty,0)$.


If you let $R = e^{-T}$ (which now takes values in $(1,\infty)$), your metric becomes $R^{-2} (dR^2 + d\Phi^2)$ and conects to the upper half plane model. From there you can invert your transformation to get to the Poincare disk model, [which has a known formula relating to the hyperboloid model](https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Relation_to_the_hyperboloid_model).


