
Journal for Geometry and Graphics 21 (2017), No. 2, 193200 Copyright Heldermann Verlag 2017 Curved Folding with Pairs of Cylinders Otto Roeschel Institute of Geometry, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria roeschel@tugraz.at 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 f_{1} and f_{2} through that curve. We examine the very special case where these two surfaces are cylinders with generators given by direction vectors e_{1} and e_{2}. 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 e_{1} and e_{2}. (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 [ Fulltextpdf (149 KB)] for subscribers only. 