Theme: SIGGRAPH Core
Folding along curves is very different from folding straight lines. The most important difference is that when you make a curved crease, the two parts of the paper that lie on opposite sides of the crease cannot be made to lie flat against each other. So while in conventional origami facets of the paper become layered and hidden, in curved origami most of the paper remains exposed to view.
This usually results in larger spans of paper that must be self supporting, so even the materials used are different: most origami paper is as thin as possible, while curved origami often requires very heavy paper or card stock.
By modeling paper as an idealized mathematical surface and applying theorems from differential geometry, we can develop methods for analyzing and designing curved origami shapes. Paper can be represented as a "developable" surface, which means that no matter how you fold or bend it, at every point on the surface it is possible to embed a straight line in the surface passing through the point. Developable surfaces may be planes, generalized cylinders or cones, or tangent surfaces.
Even for models where the curves are all circular arcs, the length and position of those arcs cannot be calculated without computer assistance. The constraints imposed to guarantee that the repeated units will tile the plane require us to numerically solve an equation containing an elliptic integral. Since the parameter solved for is not algebraic, we also need the computer to plot the resulting curves.