Recent interferometric observations by the Event Horizon Telescope have resolved the horizon-scale emission from sources in the vicinity of nearby supermassive black holes. Future space-based interferometers promise to measure the ``photon ring''---a narrow, ring-shaped, lensed feature predicted by general relativity, but not yet observed---and thereby open a new window into strong gravity. Here we present aart: an adaptive analytical ray-tracing code that exploits the integrability of light propagation in the Kerr spacetime to rapidly compute high-resolution simulated black hole images, together with the corresponding radio visibility accessible on very long space-ground baselines. The code samples images on a nonuniform adaptive grid that is specially tailored to the lensing behavior of the Kerr geometry and is therefore particularly well-suited to studying photon rings. This numerical approach guarantees that interferometric signatures are correctly computed on long baselines, and the modularity of the code allows for detailed studies of equatorial sources with complex emission profiles and time variability. To demonstrate its capabilities, we use aart to simulate a black hole movie of a stochastic, nonstationary, nonaxisymmetric equatorial source; by time-averaging the visibility amplitude of each snapshot, we are able to extract the projected diameter of the photon ring and recover the shape predicted by general relativity.