The high-velocity Ly alpha emission from SN 1987A observed with the Space Telescope Imaging Spectrograph (STIS) evidently comes from a reverse shock formed where the outer envelope of SN 1987A strikes ionized gas inside the inner circumstellar ring. The observations can be explained by a simple kinematic model, in which the Ly alpha emission comes from hydrogen atoms with radial velocity ~15,000 km s-1 crossing a reverse shock in the shape of a slightly prolate ellipsoid with equatorial radius 4.8x10^{17} cm or ~80% of the distance to the inner surface of the inner ring. N V lambda lambda 1239, 1243 emission, if present, has a net luminosity <~30% times that of the Ly alpha emission. Future STIS observations should enable us to predict the time of impact with the inner ring and to determine unambiguously whether or not N V emission is present. These observations will offer a unique opportunity to probe the structure of SN 1987A's circumstellar environment and the hydrodynamics and kinetics of very fast shocks.