Since Joseph asked for pictures, I will add to Robert's excellent response with the following images.

For the purposes of implementing Robert's change of variables in Mathematica, I simplified the composition as shown below:

```
(*twist parameter*)
t = Pi/6;
(*auxiliary function*)
g[x_, y_] := Sqrt[x^2 - y^2 + (x^2 + y^2)*Cos[2*t]];
(*change of variables for u*)
u[x_, y_] :=
Csc[t]*(Log[x^2 + y^2] -
Log[Sqrt[2]*x +
g[x, y]] + (Cos[t]*
Log[(-(x*Sqrt[1 + Cos[2*t]]) -
g[x, y])/(-(x*Sqrt[1 + Cos[2*t]]) + g[x, y])])/2);
(*parameterization*)
r[x_, y_] := {Sin[t]*Cos[u[x, y]]*y/x, Sin[t]*Sin[u[x, y]]*y/x,
Cos[t]*(Sqrt[1 - Tan[t]^2*y^2/x^2] +
Log[(Tan[t]*y/x)/(1 + Sqrt[1 - Tan[t]^2*y^2/x^2])]) +
u[x, y]*Sin[t]};
```

The domain of `r[x,y]`

is the region where $y\geq0$, $x\geq y\tan t$. This parameterization is an isometry from its domain (endowed with the Poincare half-space model metric) to its image.

First, here is an image of the parameterized surface using Mathematica's standard mesh function

```
r1 = 1; r2 = 10; ParametricPlot3D[r[x, y], {x, 0, r2}, {y, 0, r2},
RegionFunction ->
Function[{a, b, c, x, y},
x >= Tan[t]*y && r1^2 <= x^2 + y^2 <= r2^2], PlotPoints -> 100]
```

Next, let's visualize some geodesics. Below we show some geodesics in the half-plane model and their images under the parameterization.

```
(*parameter for paths*)
rad = 1;
(*Geodesics in half-plane model*)
path1[x_] := {x, Cos[t]*rad};
path2[th_] := {rad*Cos[th], rad*Sin[th]};
path3[th_] := rad*Sin[t]/Sin[2 t]*{Cos[th], Sin[th] + 1};
(*Show geodesics in half-plane model with domain*)
Show[
RegionPlot[domain, {x, 0, 10}, {y, 0, 10}, ImageSize -> Medium,
PlotStyle -> GrayLevel[0.9], BoundaryStyle -> None],
ParametricPlot[path1[x], {x, Tan[t]*rad, 10}, PlotStyle -> Red],
ParametricPlot[path2[th], {th, 0, Pi/2 - t}, PlotStyle -> Green],
ParametricPlot[path3[th], {th, -Pi/2, Pi/2 - 2 t},
PlotStyle -> Blue]
]
```

Now, those same three geodesics mapped onto Dini's Surface.

```
Show[
ParametricPlot3D[r[x, y], {x, 0, 10}, {y, 0, 10},
RegionFunction -> Function[{a, b, c, x, y}, domain],
PlotPoints -> 100, Mesh -> False, PlotStyle -> Opacity[0.5],
PlotRange -> {{-1, 1}, {-1, 1}, {-4, 4}}],
ParametricPlot3D[r @@ path1[x], {x, Tan[t]*rad, 10},
PlotStyle -> Red],
ParametricPlot3D[r @@ path2[th], {th, 0, Pi/2 - t},
PlotStyle -> Green],
ParametricPlot3D[r @@ path3[th], {th, -Pi/2, Pi/2 - 2 t},
PlotStyle -> Blue]
]
```

The blue geodesic seems to be what Joseph was imagining in the first place, following the surface to $z=-\infty$ and getting ever closer to the $z$-axis. The green geodesic leaves the rim at a right angle and follows the shortest path to the $z$-axis. Finally, most surprising to me anyway, is the red geodesic, which winds *up* the surface to $z=\infty$.