  Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the
quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all
$PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous
variety}$ if $H$ acts on it transitively, and satisfies a few other properties,
such as $H$ being semisimple. Some examples of homogeneous varieties are
quadric hypersurfaces, rational normal curves, and Veronese and Segre
embeddings. In this case, we construct new compactifications of the parameter
spaces $PGL_{N+1}/H$, obtained compactifying $PGL_{N+1}$ to the classically
known space of \textit{complete collineations}, and taking the G.I.T. quotient
by $H$, and we will call the result space of \textit{complete homogeneous
varieties}; this extends the same construction for quadric hypersurfaces in
[36]. We establish a few properties of these spaces: in particular, we find a
formula for the volume of divisors that depends only on the dimension of
$H$-invariants in irreducible representations of $SL_{N+1}$. We then develop
some tools in invariant theory, combinatorics and spline approximation to
calculate such invariants, and carry out the entire calculations for the case
of $SL_2$-invariants in irreducible representations of $SL_4$, that gives us
explicit values for the volume function in the case of $X$ being a twisted
cubic. Afterwards, we focus our attention on the case of twisted cubics, giving
a more explicit description of these compactifications, including the relation
with the previously known moduli spaces. In the end, we make some conjectures
about how the volume function might be used in solving some enumerative
problems.
