Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
Curved folding and planar cutting of simple closed curve on a conical origami
Ikhan Choi
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2016 Volume 39 Issue 3 Pages 579-595

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Abstract

The fold-and-cut theorem states that one can find a flat folding of paper, so that one complete straight cut on the folding creates any desired polygon. We extend this problem to curved origami for piecewise C1 simple closed curves. Many of those curves on paper turn out to be cut by a straight plane after we fold the paper into a conical shape—the surface consists of half-lines with a common vertex. Let γ: IR2 be a piecewise C1 simple closed curve such that there exists a parametrization γ(ψ)= (r(ψ) cos ψ, r(ψ) sin ψ) on ψ ∈ [0,2π) for a Lipschitz continuous function r: R→ (0, ∞). We prove that there exists a conical folding of the plane so that γ can be cut by a plane on the folding if a certain condition on angular total variation holds.

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© 2016 Department of Mathematics, Tokyo Institute of Technology
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