Negative Gaussian curvature from induced metric changes

We revisit the light or heat-induced changes in topography of initially flat sheets of solid that elongate or contract along patterned, in-plane director fields. For radial or azimuthal directors, negative Gaussian curvature is generated -- so-called "anti-cones". We show that azimuthal material displacements are required for the distorted state to be stretch-free and bend-minimising. The resultant shapes are smooth and aster-like and can become re-entrant in the azimuthal coordinate for large deformations. We show that care is needed when considering elastomers rather than glasses, though the former offer huge deformations.