  Using two innovations, smooth, but distinctly different, scaling laws for the
numerical reconnection of pairs of initially orthogonal and anti-parallel
quantum vortices are obtained using the three-dimensional Gross-Pitaevskii
equations, the simplest mean-field non-linear Schr\"odinger equation for a
quantum fluid. The first innovation suppresses temporal fluctuations by using
an initial density profile that is slightly below the usual two-dimensional
steady-state Pad\'e approximate profiles. The second innovation is to find the
trajectories of the quantum vortices from a pseudo-vorticity constructed on the
three-dimensional grid from the gradients of the wave function. These
trajectories then allow one to calculate the Frenet-Serret frames and the
curvature of the vortex lines. For the anti-parallel case, the scaling laws
just before and after reconnection obey the dimensional
$\delta\sim|t_r-t|^{1/2}$ prediction with temporal symmetry about the
reconnection time $t_r$ and physical space symmetry about the $x_r$, the
mid-point between the vortices, with extensions of the vortex lines formng the
edges of an equilateral pyramid. For all of the orthogonal cases, before
reconnection $\delta_{in}\sim(t-t_r)^{1/3}$ and after reconnection
$\delta_{out}\sim(t-t_r)^{2/3}$, which are respectively slower and faster than
the dimensional prediction. In these cases, the reconnection takes place in a
plane defined by the directions of the curvature and vorticity. To define the
structure further, lines are drawn that connect the four arms that extend from
the reconnection plane, four angles $\theta_i$ between these arms are found,
then summed, giving $\sum\theta_i>360^\circ$. This implies that the overall
structure is convex or hyperbolic, as opposed to the acute angles of the
anti-parallel pyramid.
