  This is the first paper in a series on event horizons in numerical
relativity. In this paper we present methods for obtaining the location of an
event horizon in a numerically generated spacetime. The location of an event
horizon is determined based on two key ideas: (1) integrating backward in time,
and (2) integrating the whole horizon surface. The accuracy and efficiency of
the methods are examined with various sample spacetimes, including both
analytic (Schwarzschild and Kerr) and numerically generated black holes. The
numerically evolved spacetimes contain highly distorted black holes, rotating
black holes, and colliding black holes. In all cases studied, our methods can
find event horizons to within a very small fraction of a grid zone.
