Inflating and programming flat inextensible curvilinear paths

Mylar balloons are popular gifts in funfairs or birthday parties.
Their conception is very simple: two pieces of flat thin sheets are cut and then sealed together along their edges. Inflation deforms the envelope as it maximises the volume of the balloon. However, while thin sheets are easy to bend or to compress (by forming wrinkles), they barely stretch, which imposes non-trivial geometrical constraints.

Here, we focus on the shape of inflated rings and, more generally, any sealed curvilinear path. We rationalise the shapes obtained for axisymmetric geometries, and in particular describe the location of wrinkles.

We find that inflation modifies the initial curvilinear flat path.
How should we choose the initial cut to obtain a mylar balloon of a desired shape? Using our theoretical predictions, we develop a simple numerical tool to solve the inverse problem of programming any 2D curve upon inflation.