  Understanding the organization of matter under the long-range electrostatic
force is a fundamental problem in multiple fields. In this work, based on the
electrically charged tethered membrane model, we reveal regular structures
underlying the lowest-energy states of inhomogeneously stretched planar
lattices by a combination of numerical simulation and analytical geometric
analysis. Specifically, we show the conformal order characterized by the
preserved bond angle in the lattice deformation, and reveal the
Poincar$\rm{\acute{e}}$-Klein mapping underlying the electrostatics-driven
inhomogeneity. The discovery of the Poincar$\rm{\acute{e}}$-Klein mapping,
which connects the Poincar$\rm{\acute{e}}$ disk and the Klein disk for the
hyperbolic plane, implies the connection of long-range electrostatic force and
hyperbolic geometry. We also discuss lattices with patterned charges of
opposite signs for modulating in-plane inhomogeneity and even creating 3D
shapes, which may have a connection to metamaterials design. This work suggests
the geometric analysis as a promising approach for elucidating the organization
of matter under the long-range force.
