  We study geometrical clues of a rigidity transition due to the emergence of a
system-spanning state of self stress in under-constrained systems of individual
polygons and spring networks constructed from such polygons. When a polygon
with harmonic bond edges and an area spring constraint is subject to an
expansive strain, we observe that convexity of the polygon is a necessary
condition for such a self stress. We prove that the cyclic configuration of the
polygon is a sufficient condition for the self stress. This correspondence of
geometry and rigidity is akin to the straightening of a one dimensional chain
of springs to rigidify it. We predict the onset of the rigidity transition
using a purely geometrical method. We also estimate the transition strain for a
given initial configuration by approximating irregular polygons as regular
polygons. These findings help determine the rigidity of an area-preserving
polygon just by looking at it. Since two-dimensional spring networks can be
considered as a network of polygons, we look for similar geometric features in
under-constrained spring networks under isotropic expansive strain. In
particular, we observe that all polygons attain convexity at the rigidity
transition such that the fraction of convex, but not cyclic, polygons predicts
the onset of the rigidity transition. Interestingly, acyclic polygons in the
network correlate with larger tensions, thus, forming effective force chains.
