Curved Folding with Pairs of Cones

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


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 \(\mathbf{c}(s)\).


Curved folding Curved origami Folding with pairs of cones 


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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of GeometryGraz University of TechnologyGrazAustria

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