  We have used the vortex filament method to numerically investigate the
interactions between pairs of quantized vortex rings that are initially
traveling in the same direction but with their axes offset by a variable impact
parameter. The interaction of two circular rings of comparable radii produce
outcomes that can be categorized into four regimes, dependent only on the
impact parameter; the two rings can either miss each other on the inside or
outside, or they can reconnect leading to final states consisting of either one
or two deformed rings. The fraction of of energy went into ring deformations
and the transverse component of velocity of the rings are analyzed for each
regime. We find that rings of very similar radius only reconnect for a very
narrow range of the impact parameter, much smaller than would be expected from
geometrical cross-section alone. In contrast, when the radii of the rings are
very different, the range of impact parameters producing a reconnection is
close to the geometrical value. A second type of interaction considered is the
collision of circular rings with a highly deformed ring. This type of
interaction appears to be a productive mechanism for creating small vortex
rings. The simulations are discussed in the context of experiments on colliding
vortex rings and quantum turbulence in superfluid helium in the zero
temperature limit.
