  The isolated horizon formalism recently introduced by Ashtekar et al. aims at
providing a quasi-local concept of a black hole in equilibrium in an otherwise
possibly dynamical spacetime. In this formalism, a hierarchy of geometrical
structures is constructed on a null hypersurface. On the other side, the 3+1
formulation of general relativity provides a powerful setting for studying the
spacetime dynamics, in particular gravitational radiation from black hole
systems. Here we revisit the kinematics and dynamics of null hypersurfaces by
making use of some 3+1 slicing of spacetime. In particular, the additional
structures induced on null hypersurfaces by the 3+1 slicing permit a natural
extension to the full spacetime of geometrical quantities defined on the null
hypersurface. This 4-dimensional point of view facilitates the link between the
null and spatial geometries. We proceed by reformulating the isolated horizon
structure in this framework. We also reformulate previous works, such as
Damour's black hole mechanics, and make the link with a previous 3+1 approach
of black hole horizon, namely the membrane paradigm. We explicit all
geometrical objects in terms of 3+1 quantities, putting a special emphasis on
the conformal 3+1 formulation. This is in particular relevant for the initial
data problem of black hole spacetimes for numerical relativity. Illustrative
examples are provided by considering various slicings of Schwarzschild and Kerr
spacetimes.
