  Intersection of a random fractal or self-affine set with a linear manifold or
another fractal set is studied, assuming that one of the sets is in a
translational motion with respect to the other. It is shown that the mass of
such an intersection is a self-affine function of the relative position of the
two sets. The corresponding Hurst exponent h is a function of the scaling
exponents of the intersecting sets. A generic expression for h is provided, and
its proof is offered for two cases --- intersection of a self-affine curve with
a line, and of two fractal sets. The analytical results are tested using
Monte-Carlo simulations.
