  The problem of characterizing the structure of an elastic network constrained
to lie on a frozen curved surface appears in many areas of science and has been
addressed by many different approaches, most notably, extending linear
elasticity or through effective defect interaction models. In this paper, we
show that the problem can be solved by considering non-linear elasticity in an
exact form without resorting to any approximation in terms of geometric
quantities. In this way, we are able to consider different effects that have
been unwieldy or not viable to include in the past, such as a finite line
tension, explicit dependence on the Poisson ratio or the determination of the
particle positions for the entire lattice. Several geometries with rotational
symmetry are solved explicitly. Comparison with linear elasticity reveals an
agreement that extends beyond its strict range of applicability. Implications
for the problem of the characterization of virus assembly are also discussed.
