  Minkowski functionals (MFs) are a set of statistics that characterise the
geometry and topology of the cosmic density field and contain complementary
information to the standard two-point analyses. We present MEDUSA, an
implementation of an accurate method for estimating the MFs of
three-dimensional point distributions. These estimates are inferred from
triangulated isodensity surfaces that are constructed from the Delaunay
tessellation of the input point sample. Contrary to previous methods, MEDUSA
can account for periodic boundary conditions, which is crucial for the analysis
of N-body simulations. We validate our code against several test samples with
known MFs, including Gaussian random fields with a ${\Lambda}$CDM power
spectrum, and find excellent agreement with the theory predictions. We use
MEDUSA to measure the MFs of synthetic galaxy catalogues constructed from
N-body simulations. Our results show clearly non-Gaussian signatures that arise
from the non-linear gravitational evolution of the density field. We find that,
although redshift-space distortions significantly change our MFs estimates,
their impact is considerably reduced if these measurements are expressed as a
function of the volume-filling fraction. We also show that the effect of
Alcock-Paczynski (AP) distortions on the MFs can be described by scaling them
with different powers of the isotropic AP parameter $q$ defined in terms of the
volume-averaged distance $D_{\rm V}(z)$. Thus the MFs estimates by MEDUSA are
useful probes of non-linearities in the density field, and the expansion and
growth of structure histories of the Universe.
