Weaving paper strips for designing of general curved surface with geometrical elasticity

TL;DR

This paper introduces 'amigami', a novel method for designing general curved surfaces by optimizing the shape of paper strips based on nonlinear elasticity theory, demonstrated through creating catenoid and helicoid forms.

Contribution

It develops a shape optimization framework for curved surfaces using geometrical elasticity and nonlinear elasticity theory on Riemannian manifolds, with numerical solutions and strain estimates.

Findings

Successfully created catenoid and helicoid surfaces with paper strips

Developed a mathematical proof for strain estimates in paper strips

Extended classical beam theory to modern geometrical elasticity

Abstract

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.