Unification postulates an anisotropic dusty obscurer around AGN, possibly a torus. It impedes view onto the central accretion engine along some lines of sight. The SEDs of such tori have been modeled extensively, but spatially resolved emission morphologies have not, presumably because the era of resolved AGN observations only now dawns (VLTI/ALMA, TMT/ELT/GMT). Recent data challenge a simple torus picture; in some objects the MIR emission is dominated by regions above the equatorial plane, suggesting that dusty outflows may be the true obscurer. A practical hurdle to systematic morphological studies is that resolved models contain ~10^5 more data than SEDs. To break this entry barrier we introduce our open-source software HYPERCAT, and with it a hypercube of AGN torus images made with CLUMPY (www.clumpy.org). This set comprises images for six model parameters and many wavelengths, and the corresponding projected dust maps. Users can also plug in their own model hypercubes. HYPERCAT interpolates these n-cubes continuously, quickly, and reliably. The software can further simulate observations of nearby AGN, both with extremely large telescopes and interferometers, including modeling the effects of PSF convolution, noise addition, detector pixelization, and image deconvolution. Using HYPERCAT we investigate AGN morphologies modeled by CLUMPY, and find that a clumpy torus can produce N-band emission patterns extended in polar direction, compatible with many observations. We further determine the resolvability of nearby AGN with extremely large telescopes. We release HYPERCAT, hopeful that it will facilitate studies of resolved AGN morphologies without the worry of handling very large sets of model images.