Mathematical Approach to Model Foldable Conical Structures Using Conformal Mapping

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. To the best of our knowledge, this study is the first attempt to disclose a comprehensive design approach that can simplify the conventional design process. The proposed design approach can be addressed for further foldable structures such as circular membranes and toroidal tubes to broaden the design possibility of foldable mechanical products.