Gluing and Creasing Paper along Curves: Computational Methods for Analysis and Design

Abstract

Curved geometries that can be obtained from flat sheets of material have many potential applications in design and engineering. In this thesis, we consider shapes achievable by joining planar patches of material along their curved boundaries, focusing specifically on curved-crease origami as a special case. Our research makes a threefold contribution.

First, we extend the theory behind the computation of shapes consisting of developable patches. Based on classical differential geometry using curvature-based analysis, we consider the gluing of two patches with specified rulings and the pairwise joining of three patches with partial ruling information. We highlight a simplified computational approach for the case when the joined two patches are cylinders or cones that is also applicable in the discrete case. Additionally, we show how to compute a crease that connects a patch with a patch that is composed of tangent-continuous cylinders and cones.

Using this theory, we are able to extend the family of shapes that allow a parametric reconstruction. We provide examples of shapes that allow explicit parametrizations, parametrizations using elliptic integrals, and parametrizations that require numerical integration.

Finally, we employ the developed theory to devise algorithmic design strategies for shapes with curved creases. Inspired by artistic origami, we offer a parametric design tool for the construction of origami spirals. Additionally, we consider two strategies that approximate a polyhedral shape with a modular curved-crease design. Finally, we provide a constructive linear subdivision scheme for regular developable planar quad meshes that correspond to a discretized curved-crease shape.

With this thesis, we aim to make curved-crease origami more accessible for interdisciplinary research in various design and engineering contexts.