  Origami describes rules for creating folded structures from patterns on a
flat sheet, but does not prescribe how patterns can be designed to fit target
shapes. Here, starting from the simplest periodic origami pattern that yields
one degree-of-freedom collapsible structures, we show that scale-independent
elementary geometric constructions and constrained optimization algorithms can
be used to determine spatially modulated patterns that yield approximations to
given surfaces of constant or varying curvature. Paper models confirm the
feasibility of our calculations. We also assess the difficulty of realizing
these geometric structures by quantifying the energetic barrier that separates
the metastable flat and folded states. Moreover, we characterize the trade-off
between the accuracy to which the pattern conforms to the target surface, and
the effort associated with creating finer folds. Our approach enables the
tailoring of origami patterns to drape complex surfaces independent of absolute
scale, and quantify the energetic and material cost of doing so.
