Curved folding and planar cutting of simple closed curve on a conical origami

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: I → R² 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 $\in$ [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.