  The complex morphologies exhibited by spatially confined thin objects have
long challenged human efforts to understand and manipulate them, from the
representation of patterns in draped fabric in Renaissance art to current day
efforts to engineer flexible sensors that conform to the human body. We
introduce a theoretical principle, broadly generalizing Euler's
{\emph{elastica}} -- a core concept of continuum mechanics that invokes the
energetic preference of bending over straining a thin solid object and has been
widely applied to classical and modern studies of beams and rods. We define a
class of {\emph{geometrically incompatible confinement}} problems, whereby the
topography imposed on a thin solid body is incompatible with its intrinsic
("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}},
induces strain. Focusing on a prototypical example of a sheet attached to a
spherical substrate, numerical simulations and analytical study demonstrate
that the mechanics is governed by a principle, which we call the "Gauss-Euler
{\emph{elastica}}". This emergent rule states that - despite the unavoidable
strain in such an incompatible confinement - the ratio between the energies
stored in straining and bending the solid may be arbitrarily small. The
Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly
simplifies the daunting task of solving the highly nonlinear equations that
describe thin solids at mechanical equilibrium. This development thus opens new
possibilities for attacking a broad class of phenomena governed by the coupling
of geometry and mechanics.
