We present a computational framework for interactive design and exploration of curved folded surfaces. In current practice, such surfaces are typically created manually using physical paper, and hence our objective is to lay the foundations for the digitalization of curved folded surface design. Our main contribution is a discrete binary characterization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our machinery to build the first interactive freeform editing tool capable of modeling bending and folding of complicated crease patterns.
The authors would like to thank Oliver Glauser and Justin Solomon for illuminating discussions, Martin Kilian and Niloy Mitra for clarifications about their prior work, and Katja Wolff and Phillip Herholz for their help with results, figures and video production. The work was supported in part by the Deutsche Forschungsgemeinschaft-Collaborative Research Center, TRR 109, "Discretization in Geometry and Dynamics" and gifts from Facebook, Adobe and Snap, Inc.