  We study the metric dimension and optimal split-resolving sets of the
point-circle incidence graph of a M\"obius plane. We prove that the metric
dimension of a M\"obius plane of order $q$ is around $2q$, and that an optimal
split-resolving set has cardinality between approximately $5q$ and $2.5q\log
q$. We also prove that a smallest blocking set of a M\"obius plane of order $q$
has at most $2q(1 + \log(q + 1))$ points.
