9/3/2014 Authors: Sachiko Ishida, Taketoshi Nojima and Ichiro Hagiwara
J. Mech. Des. 136(9), 091007 (2014) 
doi: 10.1115/1.4027848

A new approach for obtaining the crease patterns of foldable conical structures from crease patterns of cylindrical structures based on the origami folding theory using conformal mapping is presented in this paper. Mapping for flow with circulation, which is the so-called polar conversion, is demonstrated as an example. This mapping can be used to produce similar elements and maintain the regularity of fold lines. This is a significant advantage when the mapping approach is used to produce foldable structures, because

it is relatively easy to control angles between fold lines. Thus, this proposed approach enables us to design complex structures from simple original structures systematically, maintaining advanced characteristics particular to origami such as folding up spatial structures onto a plane and expanding them at will.

Crease patterns and physical model; (a): original crease pattern of cylindrical structure; (b): transformed crease pattern of conical structure after angle correction; (c): physical model of conical structure. Extracted from Fig. 4

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