  Geometrical structures intrinsic to non-expanding, weakly isolated and
isolated horizons are analyzed and compared with structures which arise in
other contexts within general relativity, e.g., at null infinity. In
particular, we address in detail the issue of singling out the preferred
normals to these horizons required in various applications. This work provides
powerful tools to extract invariant, physical information from numerical
simulations of the near horizon, strong field geometry. While it complements
the previous analysis of laws governing the mechanics of weakly isolated
horizons, prior knowledge of those results is not assumed.
