  This study proposes 'amigami' as a new method of creating a general curved
surface. It conducts the shape optimization of weaving paper strips based on
the theory of nonlinear elasticity on Riemannian manifolds. The target surface
is split into small curved strips by cutting the medium along with its
coordinates, and each strip is embedded into a flat paper sheet to minimize a
strain energy functional due to the in-plane deformation. The weak form
equilibrium equation is derived from a Lie derivative with the virtual
displacement vector field, and the equation is solved numerically using the
Galerkin method with a non-uniform B-spline manifold. As a demonstration, we
made catenoid and helicoid surfaces which are made by waving 54 paper strips.
The papercraft reminds us of the isometric transformation from the catenoid to
the helicoid and vice versa. We also provide strain estimates for paper strips
with rigorous mathematical proof. This estimating process is a generalization
of the classical beam theory of Euler-Bernoulli to a modern geometrical
elasticity.
