  We study a model of programmable matter systems consisting of $n$ devices
lying on a 2-dimensional square grid, which are able to perform the minimal
mechanical operation of rotating around each other. The goal is to transform an
initial shape A into a target shape B. We are interested in characterising the
class of shapes which can be transformed into each other in such a scenario,
under the additional constraint of maintaining global connectivity at all
times. This was one of the main problems left open by $[$Michail et al.,
JCSS'19$]$. Note that the considered question is about structural feasibility
of transformations, which we exclusively deal with via centralised constructive
proofs. Distributed solutions are left for future work and form an interesting
research direction. Past work made some progress for the special class of nice
shapes. We here consider the class of orthogonal convex shapes, where for any
two nodes $u, v$ in a horizontal or vertical line on the grid, there is no
empty cell between $u$ and $v$. We develop a generic centralised transformation
and prove that, for any pair $A$, $B$ of colour-consistent orthogonal convex
shapes, it can transform $A$ into $B$. In light of the existence of blocked
shapes in the considered class, we use a minimal 3-node seed to trigger the
transformation. The running time of our transformation is an optimal $O(n^2)$
sequential moves, where $n=|A|=|B|$. We leave as an open problem the existence
of a universal connectivity-preserving transformation with a small seed. Our
belief is that the techniques developed in this paper might prove useful to
answer this.
