A tractrix in the

plane, whose asymptote is the
axis and which intersects the
axis at a unit distance from the origin, can be parametrized as
,
, where
. Rotating this curve around the
axis, we end up with the pseudosphere, whose first fundamental form is
; this relation introduces a metric in the

halfplane, which coincides with the metric in Poincaré's halfplane model.
The
coordinate of a point on the pseudosphere represents the angle of rotation around the asymptote of the tractrix. The parametrization has period
in
and every point on the pseudosphere corresponds to infinitely many points in the

plane. Consequently, there are infinitely many geodesics joining a given pair of points on the pseudosphere.
For more details, see Example 9.3.3 and Section 11.1 in [1]. The role of the pseudosphere in the history of hyperbolic geometry is described in [2].
Making the surface semitransparent gives a better overview of the geodesic behavior, but takes longer to render.
[1] A. Pressley,
Elementary Differential Geometry, 2nd ed., London: Springer, 2010.
[2] J. Stillwell,
Mathematics and Its History, 3rd ed., London: Springer, 2010.