So now our friends have found their keys, and informed each other of
that fact [unless of course they chose 
in which case
they can't find each other now!]. So it is time to head for the
country for a walk in the bush to clear their heads, after that
tiresome search. During their walk they become separated and although
they can ascertain the distance between them, they can't find each
other. Fortuitously, they both were brushing up on their geometry
during the ride and are familiar with the method supplied to them by
Fakete([3]).
Figure 4: The
Kissing Circles algorithm, with r=0
By following the solid lines in Figure 4 they are assured of doing no better at finder one another with their eyes closed, so to speak. This is assuming that their compasses are perfectly calibrated and the Earth being flat [a fact which we know not to be true, see [5]]. One of the friends initially walks south, and the other north, then they follow a circle about their starting positions until they bump into one another [See Fakete [3] for a full explanation].
By introducing disparity in the compass readings this algorithm will not succeed. Provided our agents open their eyes and so can see about themselves a radius, r>0, we conjecture that the following modification to the Kissing Circle algorithm will maximize their chances of reuniting. It goes as follows.
Figure 5: The modified Kissing Circles algorithm, with r>0
Again, one walks south and the other north, according to their
personal compasses. Now, however, it is conjectured that if they walk
the same distance they walked before plus half the distance they can
sense, 
, and
follow the circle etched out by that radius then they will find each
other somewhere in the arc spanned by in figure
5. We believe this method maximizes the discovery
arc-length.