Geometry for evolving topographies of light-responsive plastic sheets

Recently, topography change by illumination of pre-stretched, flat sheets covered in ink of optical density varying in-plane has been demonstrated by Mailen\textit{ et al}, Smart Materials and Structures, 2019. They reduce an analysis of the problem to one of metric change in the sheets in the thin limit, that is, to a question of geometry. We present the explicit form of the contraction field needed to produce the bowls these authors were interested in, using a method that can also yield the contraction field for more general desired, circularly-symmetric topography development. We give as examples the fields required for developing paraboloids and catenoids.