  We study the intersection points of a fixed planar curve $\Gamma$ with the
nodal set of a translationally invariant and isotropic Gaussian random field
$\Psi(\bi{r})$ and the zeros of its normal derivative across the curve. The
intersection points form a discrete random process which is the object of this
study. The field probability distribution function is completely specified by
the correlation $G(|\bi{r}-\bi{r}'|) = <\Psi(\bi{r}) \Psi(\bi{r}')>$.
  Given an arbitrary $G(|\bi{r}-\bi{r}'|)$, we compute the two point
correlation function of the point process on the line, and derive other
statistical measures (repulsion, rigidity) which characterize the short and
long range correlations of the intersection points. We use these statistical
measures to quantitatively characterize the complex patterns displayed by
various kinds of nodal networks. We apply these statistics in particular to
nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of
special interest is the observation that for monochromatic random waves, the
number variance of the intersections with long straight segments grows like $L
\ln L$, as opposed to the linear growth predicted by the percolation model,
which was successfully used to predict other long range nodal properties of
that field.
