For a $d$-dimensional simplicial complex $\Delta \subset {\RR}^d$
such that $\Delta$ and all its links are pseudomanifolds, we
consider the module $\asp$ of mixed splines. In particular,
we study the freeness of the module $\hasp$ for a triangulation
$\Delta \subset {\RR}^2$ of a topological disk and for a
non-negative integer vector $\alpha$ of length $f_1^0 (\Delta)$,
where $\widehat \Delta \subset {\RR}^3$ is the join of $\Delta$
with the origin in ${\RR}^3$ and $f_1^0 (\Delta)$ is the number
of interior edges in $\Delta$.
We completely characterize $\Delta$ for which $\hasp$ is free
for any non-negative integer vector $\alpha$.
Moreover, we obtain a method for determining whether $\hasp$ is
free for a triangulation $\Delta \subset {\RR}^2$ of a topological
disk which has a totally interior edge, and for a generic
non-negative integer vector $\alpha$.