  Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is
there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The
answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be
convex, connected, or disconnected. We show that our result can be generalised
to give an interpolated shape between $A$ and $B$ for any interpolation
variable $\alpha$ between $0$ and $1$, and prove that the resulting morph has a
bounded rate of change with respect to $\alpha$. Finally, we explore a
generalization of the concept of a Hausdorff middle to more than two input
sets. We show how to approximate or compute this middle shape, and that the
properties relating to the connectedness of the Hausdorff middle extend from
the case with two input sets. We also give bounds on the Hausdorff distance
between the middle set and the input.
