On the Rigid-Ruling Folding of Curved Creases: Conjugate-Net Preserving Isometric Deformations of Semi-Discrete Globally Developable Conjugate-Nets

TL;DR

This paper studies the mathematical conditions for rigid folding of curved crease patterns, introducing methods for constructing foldable patterns and analyzing the compatibility of different crease types.

Contribution

It provides new conditions for foldability, methods for constructing foldable patterns, and characterizes compatible crease combinations in semi-discrete developable conjugate nets.

Findings

Derived conditions for rigid-ruling foldability of curve pairs

Developed computational methods for constructing foldable crease patterns

Characterized compatibility of planar and constant fold-angle creases

Abstract

In this paper, we investigate rigid-ruling folding motions of crease-rule patterns, that is, conjugacy-preserving isometries of developable semi-discrete conjugate nets. We derive two conditions for the rigid-ruling foldability of pairs of curves and consider two applications. First, we introduce computations that enable the sequential construction of rigid-ruling foldable crease-rule patterns. Second, we examine combinations of planar and constant fold-angle creases. In particular, we show that constant fold-angle creases are only compatible with other constant fold-angle creases, and we provide a characterization of rigid-ruling foldable combinations of planar and constant fold-angle creases.