  We present a novel approach to understand geometric-incompatibility-induced
rigidity in under-constrained materials, including sub-isostatic 2D spring
networks and 2D and 3D vertex models for dense biological tissues. We show that
in all these models a geometric criterion, represented by a minimal length
$\bar\ell_\mathrm{min}$, determines the onset of prestresses and rigidity. This
allows us to predict not only the correct scalings for the elastic material
properties, but also the precise {\em magnitudes} for bulk modulus and shear
modulus discontinuities at the rigidity transition as well as the magnitude of
the Poynting effect. We also predict from first principles that the ratio of
the excess shear modulus to the shear stress should be inversely proportional
to the critical strain with a prefactor of three, and propose that this factor
of three is a general hallmark of geometrically induced rigidity in
under-constrained materials and could be used to distinguish this effect from
nonlinear mechanics of single components in experiments. Lastly, our results
may lay important foundations for ways to estimate $\bar\ell_\mathrm{min}$ from
measurements of local geometric structure, and thus help develop methods to
characterize large-scale mechanical properties from imaging data.
