  The asymmetric shape of the nebula around $\eta$-Carinae (Homunculus) can be
explained by a spherical expansion in a non-homogeneous medium. Two models are
analyzed: an exponential and an inverse power law dependence for the density as
a function of distance from the equatorial plane. The presence of a medium with
variable density along the polar direction progressively converts the original
spherical shell into a bipolar nebula. In the case of the nebula around
$\eta$-Carinae, we know the time elapsed since the great outburst in 1840. An
exact match between observed radii and velocities can be obtained by fine
tuning the parameters involved, such as initial radius, initial velocity and
the typical scale that characterizes the gradient in density. The observed
radius and velocity of the Homunculus as a function of the polar angle in
spherical coordinates can be compared with the corresponding simulated data by
introducing the efficiency in a single or multiple directions. Once the 3D
spatial structure of the Homunculus is obtained, we can compose the image by
integrating along the line of sight. In order to simulate the observed image,
we have considered a bipolar nebula with constant thickness and an optically
thin emitting layer. Some simulated cuts of the relative intensity are reported
and may represent a useful reference for the astronomical cuts.
