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We solve the forward and inverse problems associated with the transformationof flat sheets to surfaces of revolution with non-trivial topography, includingGaussian curvature, without a stretch elastic cost. We deal with systemsslender enough to have a small bend energy cost. Shape change is induced bylight or heat causing contraction along a non-uniform director field in theplane of an initially flat nematic sheet. The forward problem is, given adirector distribution, what shape is induced? Along the way, we determine theGaussian curvature and the evolution with induced mechanical deformation of thedirector field and of material curves in the surface (proto-radii) that willbecome radii in the final surface. The inverse problem is, given a targetshape, what director field does one need to specify? Analytic examples ofdirector fields are fully calculated that will, for specific deformations,yield catenoids and paraboloids of revolution. The general prescription isgiven in terms of an integral equation.

Nematic director fields and topographies of shells of revolution