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

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.

Figures (5)

• Figure 1: Illustration of concepts discussed in Section \ref{['sec:intro']}.
• Figure 2: Illustration of the notation of a developable patch and its development (as in mundilovaphdmundilova2024planar).
• Figure 3: Illustration of the notation introduced in Section \ref{['sec:singlecrease']} (as in mundilovaphdmundilova2024planar).
• Figure 4: Illustration of the notation introduced in Section \ref{['sec:multiplecrease']} (as in mundilovaphdmundilova2024planar).
• Figure 5: Illustration of a rigid-ruling folding motion of a crease-rule pattern with tangent-parallel creases mundilovaphd.

Theorems & Definitions (21)

• Lemma 1
• Definition 1
• Definition 2
• Definition 3
• Lemma 2
• Lemma 3
• proof
• Corollary 1
• Lemma 4
• Lemma 5