  We present approximate analytical solutions to the Hamiltonian and momentum
constraint equations, corresponding to systems composed of two black holes with
arbitrary linear and angular momentum. The analytical nature of these initial
data solutions makes them easier to implement in numerical evolutions than the
traditional numerical approach of solving the elliptic equations derived from
the Einstein constraints. Although in general the problem of setting up initial
conditions for black hole binary simulations is complicated by the presence of
singularities, we show that the methods presented in this work provide initial
data with $l_1$ and $l_\infty$ norms of violation of the constraint equations
falling below those of the truncation error (residual error due to
discretization) present in finite difference codes for the range of grid
resolutions currently used. Thus, these data sets are suitable for use in
evolution codes. Detailed results are presented for the case of a head-on
collision of two equal-mass M black holes with specific angular momentum 0.5M
at an initial separation of 10M. A straightforward superposition method yields
data adequate for resolutions of $h=M/4$, and an "attenuated" superposition
yields data usable to resolutions at least as fine as $h=M/8$. In addition, the
attenuated approximate data may be more tractable in a full (computational)
exact solution to the initial value problem.
