  We resolve a conjecture of F\"assler and Orponen on the dimension of
exceptional projections to one-dimensional subspaces indexed by a space curve
in $\mathbb{R}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of
the Wolff circular maximal function over fractal sets for a class of $C^2$
curves related to Sogge's cinematic curvature condition. A key new tool is the
use of lens cutting techniques from discrete geometry.
