# Curved Folding with Pairs of Cones

• Otto Röschel
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)

## Abstract

On a sheet of paper we consider a curve $$\mathbf{c}^*(s)$$. ‘Curved paper folding’ (or ‘curved Origami’) along $$\mathbf{c}^*(s)$$ folded from the planar sheet yields a (spatial) curve $$\mathbf{c}(s)$$ while the paper on both sides of $$\mathbf{c}(s)$$ turns into two developable strips $$\varPhi _{1,2}$$ through that curve. We examine the very special case of a configuration where the two surfaces $$\varPhi _ {1,2}$$ happen to be cones with different vertices $$\mathbf{v}_{1,2}$$. Such a triple $$(\mathbf{c}(s), \mathbf{v}_{1}, \mathbf{v}_{2})$$ shall be termed ‘triple for curved folding with pairs of cones’. In this paper we prove the following characterization of such triples: $$(\mathbf{c}(s), \mathbf{v}_{1}, \mathbf{v}_{2})$$, in general, is a triple for curved folding with cones iff the developable surface enveloped by the osculating planes of $$\mathbf{c}(s)$$ is tangent to some quadric of revolution $$\varPsi$$ with two different real focal points on its axis of rotation. These real focal points of $$\varPsi$$ are the vertices $$\mathbf{v}_{1,2}$$ of the two cones. If the curve $$\mathbf{c}(s)$$ happens to be planar we arrive at one of the following special cases: The two vertices $$\mathbf{v}_{1,2}$$ have to be symmetric with respect to the plane of $$\mathbf{c}(s)$$, or both, $$\mathbf{v}_{1}$$ and $$\mathbf{v}_{2}$$, are contained in the plane of $$\mathbf{c}(s)$$.

## Keywords

Curved folding Curved origami Folding with pairs of cones

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