Discretizations of Conic Crease Patterns

Joint work with Erik Demaine and Tomohiro Tachi.

A locally flat and rigidly foldable discretization of one of
David Huffman’s designs with scaled and reflected parabolas.

Conic curved creases with reflected rule lines is a style of curved origami design, first explored by David Huffman, that is attractive in that it gives one-DOF folding motions with rigid rule lines (i.e., the rule lines remain the same throughout the motion). We show how to discretize any such curved crease pattern into a similar straight-line crease pattern that has a one-DOF rigid folding motion. We develop two general methods for such discretization, where each curve is replaced by an inscribing or circumscribing polygonal line, respectively, and show in both cases that the resulting discretized crease patterns are rigidly foldable. In the case of the circumscribed discretization, the crease pattern is also locally flat foldable. On the other hand, only careful sampling in the inscribed method results in locally flat-foldable crease patterns.

Read more in our paper.

Rigid ruling folding of a conical crease pattern consisting of confocal ellipses.
Locally flat and rigidly foldable circumscribed discretization of a crease pattern consisting of confocal ellipses.
Rigidly foldable inscribed discretization of a crease pattern
consisting of confocal ellipses.

We implemented a Grasshopper / Rhino plug-in for experimental design. Here are some further examples.

Locally flat and rigidly foldable discretization of a variation of a design by Huffman consisting of ellipses and hyperbolas.
Rigidly foldable design with parabolic creases.
Rigidly foldable spiral with parabolic creases.
Discretizations of variations of Huffman’s ‘4-lobed cloverleaf’ design.