Global inverse design of anisotropiclly deforming sheets using curve defects

Can an anisotropically-deforming, shape-shifting sheet transform into any shape? While locally the local answer is positive, the global answer, so far, has been no. Smooth solutions to the inverse design problem develop singularities within a finite distance, preventing them from covering as simple a surface as a sphere. 

Here we show how the introduction of curve defects alleviates this problem. Demanding that these lines are invisible upon actuation, i.e. carry no gaussian curvature, uniquely determines the smooth director field across the curve, while allowing it to expand beyond the singularity.