  We consider the optimal covering of fractal sets in a two-dimensional space
using ellipses which become increasingly anisotropic as their size is reduced.
If the semi-minor axis is \epsilon and the semi-major axis is \delta, we set
\delta=\epsilon^\alpha, where 0<\alpha<1 is an exponent characterising the
anisotropy of the covers. For point set fractals, in most cases we find that
the number of points N which can be covered by an ellipse centred on any given
point has expectation value < N > ~ \epsilon^\beta, where \beta is a
generalised dimension. We investigate the function \beta(\alpha) numerically
for various sets, showing that it may be different for sets which have the same
fractal dimension.
