Is there a special case in which wave atoms convert to curvelets?
In their 2007 paper, Wave atoms and sparsity of oscillatory patterns, Demanet and Ying  draw connections between wave atoms and other types of wavelets:



In plain words:


  We introduce “wave atoms” as a variant of 2D wavelet packets obeying
  the parabolic scaling wavelength


with a better spatial localization, meant for texture analysis. Their choice ($\alpha=\beta=1/2$) was a compromise between two types of sparsification (preservation under warping, sparsity of oscillations). Yet, one could built different  collections of such wave packets, with alternative  choices of  $\alpha$ and $\beta$, bridging the gap between the different transform choices.

Yet, their natural redundancy is 2,  with an orthonormal variant, and a complex one with redundancy 4, different from that of the curvelets. So I do no think they can be used as a special case of curvelets. 

A tutorial on 2D geometric transformation can be obtain in A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity, 2011.