Rigid folding representation by the Stereographic Projection

•Representation of rigid facets rotating around a corner of an origami-like structure•Stereographic Projection of the Riemann sphere on the Gauss plane•Nodal points, nodal line, conjugate circles, pole. Trajectories and rays•Basic problems. Folding about a generic crease•Chain of the successive rotation about different creases.

The Stereographic Projection machinery is applied to the qualitative and quantitative description of the folding problem of an origami-like structure: the facets of a creased rigid sheet undergo successive rotation about several creases arbitrarily located in the three-dimensional space. The theoretical apparatus is revisited and some basic problems are depicted to emphasize both the simplicity and the great efficiency of the representation. Several original results, useful to simplify the trajectory tracing of the facet, are derived. The Representation of the Pole Path around a corner is finally presented as an alternative synthetic description of the folded configuration. An application to a multi-folded carton erection demonstrates the effectiveness of the proposed formulation.

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