  In the classic Symmetric Rendezvous problem on a Line (SRL), two robots at
known distance 2 but unknown direction execute the same randomized algorithm
trying to minimize the expected rendezvous time. A long standing conjecture is
that the best possible rendezvous time is 4.25 with known upper and lower
bounds being very close to that value. We introduce and study a geometric
variation of SRL that we call Symmetric Rendezvous in a Disk (SRD) where two
robots at distance 2 have a common reference point at distance $\rho$. We show
that even when $\rho$ is not too small, the two robots can meet in expected
time that is less than $4.25$. Part of our contribution is that we demonstrate
how to adjust known, even simple and provably non-optimal, algorithms for SRL,
effectively improving their performance in the presence of a reference point.
Special to our algorithms for SRD is that, unlike in SRL, for every fixed
$\rho$ the worst case distance traveled, i.e. energy that is used, in our
algorithms is finite. In particular, we show that the energy of our algorithms
is $O\left(\rho^2\right)$, while we also explore time-energy tradeoffs,
concluding that one may be efficient both with respect to time and energy, with
only a minor compromise on the optimal termination time.
