Measuring the shape of a black hole photon ring

General relativity predicts that gravitational lensing near black holes will produce narrow "photon rings" on images. Building on recent work of Johnson, Lupsasca et al. focusing on circular rings, I calculate the long-baseline interferometric signature of a narrow feature of arbitrary shape and intensity. The shape information is contained in the oscillation frequencies of the complex visibility as a function of baseline length, for each baseline angle. These results raise the possibility of measuring the detailed shape of a photon ring and comparing to the precise predictions of general relativity.