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Journal for Geometry and Graphics 21 (2017), No. 2, 193--200
Copyright Heldermann Verlag 2017

Curved Folding with Pairs of Cylinders

Otto Roeschel
Institute of Geometry, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria

On a sheet of paper we consider a curve c*(s). 'Curved paper folding' or 'curved Origami' along c*(s) folded from the planar sheet yields a spatial curve c(s) and two developable strips f1 and f2 through that curve. We examine the very special case where these two surfaces are cylinders with generators given by direction vectors e1 and e2. In this paper we prove the following properties and statements:
(a) The spherical image c'(s) of the tangent vectors of c(s) is, in general, contained in a spherical conic with two of its foci in the directions of e1 and e2.
(b) Any possible curve c(s) is affinely related to a curve of constant slope.
The results are also transferred to the discrete case where c(s) is replaced by a spatial polygon while the cylinders turn into prisms.

Keywords: Curved origami, curved folding, geodesic curvature, origami with pairs of cylinders.

MSC: 53A05; 51N05, 68U07

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