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.