Leaves display a remarkable range of forms, from flat sheets with simple outlines to cup-shaped traps. Although much progress has been made in understanding the mechanisms of planar leaf development, it is unclear whether similar or distinctive mechanisms underlie shape transformations during development of more complex curved forms. Here, we use 3D imaging and cellular and clonal analysis, combined with computational modelling, to analyse the development of cup-shaped traps of the carnivorous plant Utricularia gibba. We show that the transformation from a near-spherical form at early developmental stages to an oblate spheroid with a straightened ventral midline in the mature form can be accounted for by spatial variations in rates and orientations of growth. Different hypotheses regarding spatiotemporal control predict distinct patterns of cell shape and size, which were tested experimentally by quantifying cellular and clonal anisotropy. We propose that orientations of growth are specified by a proximodistal polarity field, similar to that hypothesised to account for Arabidopsis leaf development, except that in Utricularia, the field propagates through a highly curved tissue sheet. Independent evidence for the polarity field is provided by the orientation of glandular hairs on the inner surface of the trap. Taken together, our results show that morphogenesis of complex 3D leaf shapes can be accounted for by similar mechanisms to those for planar leaves, suggesting that simple modulations of a common growth framework underlie the shaping of a diverse range of morphologies.