Molding 3D curved structures by selective heating

It is of interest to fabricate curved surfaces in three dimensions from
homogeneous material in the form of flat sheets. The aim is not just to
obtain a surface which has a desired intrinsic Riemannian metric, but to
get the desired embedding in R^3 up to translations and rotations. In this
paper, we demonstrate three generic methods of molding a flat sheet of
thermo-responsive plastic by selective contraction induced by targeted
heating. These methods do not involve any cutting and gluing, which is a
property they share with origami. The first method is inspired by
tailoring, which is the usual method for making garments out of plain
pieces of cloth. Unlike usual tailoring, this method produces the desired
embedding in R^3. The second method just aims to bring about the desired
new Riemannian metric via an appropriate pattern of local contractions,
without directly controlling the embedding. The third method is based on
triangulation, and seeks to induce the desired local distances. This
results in getting the desired embedding in R^3. The second and the third
methods, and also the first method for the special case of surfaces of
revolution, are algorithmic in nature. We explain these methods, and show
examples.