Designing Freeform Origami Tessellations by Generalizing Resch's Patterns

In this research, we study a method to produce families of origami tessellations from given polyhedral surfaces. The resulting tessellated surfaces generalize the patterns proposed by Ron Resch and allow the construction of an origami tessellation that approximates a given surface. We will achieve these patterns by first constructing an initial configuration of the tessellated surfaces by separating each facets and inserting folded parts between them based on the local configuration. The initial configuration is then modified by solving the vertex coordinates to satisfy geometric constraints of developability, folding angle limitation, and local nonintersection. We propose a novel robust method for avoiding intersections between facets sharing vertices. Such generated polyhedral surfaces are not only applied to folding paper but also sheets of metal that does not allow 180 deg folding.