  This paper contains new results on two classical topics in fractal geometry:
projections, and intersections with affine planes. To keep the notation of the
abstract simple, we restrict the discussion to the planar cases of our
theorems.
  Our first main result considers the orthogonal projections of two Borel sets
$A,B \subset \mathbb{R}^{2}$ into one-dimensional subspaces. Under the
assumptions $\dim A \leq 1 < \dim B$ and $\dim A + \dim B > 2$, we prove that
the intersection of the projections $P_{L}(A)$ and $P_{L}(B)$ has dimension at
least $\dim A - \epsilon$ for positively many lines $L$, and for any $\epsilon
> 0$. This is quite sharp: given $s,t \in [0,2]$ with $s + t = 2$, we construct
compact sets $A,B \subset \mathbb{R}^{2}$ with $\dim A = s$ and $\dim B = t$
such that almost all intersections $P_{L}(A) \cap P_{L}(B)$ are empty. In case
both $\dim A > 1$ and $\dim B > 1$, we prove that the intersections $P_{L}(A)
\cap P_{L}(B)$ have positive length for positively many $L$.
  If $A \subset \mathbb{R}^{2}$ is a Borel set with $0 < \mathcal{H}^{s}(A) <
\infty$ for some $s > 1$, it is known that $A$ is 'visible' from almost all
points $x \in \mathbb{R}^{2}$ in the sense that $A$ intersects a positive
fraction of all lines passing through $x$. In fact, a result of Marstrand says
that such non-empty intersections typically have dimension $s - 1$. Our second
main result strengthens this by showing that the set of exceptional points $x
\in \mathbb{R}^{2}$, for which Marstrand's assertion fails, has Hausdorff
dimension at most one.
