By using black hole telescopes such as the EHT (Event Horizon Telescopes) on relatively nearby black holes, we can now construct images of black holes, and from them extract certain key features. One such feature is the "photon ring" orbiting the black hole, which is really an infinite sequence of self-similar rings of photons getting asymptotically thinner, which are formed by the "nearly bound" geodesic trajectories of photons near the black hole's bound-orbit radius. The precise shape of these rings, and their relation to one another (as viewed from Earth), is determined by the metric of spacetime around the black hole, which depends on certain parameters of the black hole, such as mass, charge, and spin. In this project, I plan to use the equations of general relativity to derive, either analytically or computationally, several characteristic quantities of these photon rings (namely, the 3 "critical exponents") for black holes of a given metric. My final product, besides the Senior Project blog and presentation, will likely be a paper detailing the full derivations as well as providing graphs of the critical exponent as a function of polar angle in the image, which would provide insight into how strongly the critical exponents depend on the polar angle as well as on the black hole parameters.