  We revisit the geometrical meaning of statistical isotropy that is manifest
in excursion sets of smooth random fields in two dimensions. Using the contour
Minkowski tensor, $\W_1$, as our basic tool we first examine geometrical
properties of single structures. For simple closed curves in two dimensions we
show that $\W_1$ is proportional to the identity matrix if the curve has
$m$-fold symmetry, with $m\ge 3$. Then we elaborate on how $\W_1$ maps any
arbitrary shaped simple closed curve to an ellipse that is unique up to
translations of its centroid. We also carry out a comparison of the shape
parameters, $\alpha$ and $\beta$, defined using $\W_1$, with the filamentarity
parameter defined using two scalar Minkowski functionals - area and contour
length. We show that they contain complementary shape information, with $\W_1$
containing additional information of orientation of structures. Next, we apply
our method to boundaries of excursion sets of random fields and examine what
statistical isotropy means for the geometry of the excursion sets. Focusing on
Gaussian isotropic fields, and using a semi-numerical approach we quantify the
effect of finite sampling of the field on the geometry of the excursion sets.
In doing so we obtain an analytic expression for $\alpha$ which takes into
account the effect of finite sampling. Finally we derive an analytic expression
for the ensemble expectation of $\W_1$ for Gaussian anisotropic random fields.
Our results provide insights that are useful for designing tests of statistical
isotropy using cosmological data.
