  We show that the maximum number of pairwise non-overlapping $k$-rich lenses
(lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the
plane is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$,
and the sum of the degrees of the lenses of such a family (where the degree of
a lens is the number of circles that form it) is
$O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{3/2}} + n\right)$. Two independent
proofs of these bounds are given, each interesting in its own right (so we
believe). We then show that these bounds lead to the known bound of Agarwal et
al. (JACM 2004) and Marcus and Tardos (JCTA 2006) on the number of point-circle
incidences in the plane. Extensions to families of more general algebraic
curves and some other related problems are also considered.
