  We present an extension to $\pi^{\pm} p$ and $K^{\pm} p$ elastic scattering
at high energies of the impact-picture phenomenology we first proposed more
than twenty years ago, for $p p$ and $\bar p p$ elastic scattering. We show, in
particular, that the analytic form of the opacity function for the proton
obtained previously is compatible with the experimental results on $\pi p$ and
$K p$ elastic scattering at high energies. It is proposed that $\pi^{\pm}$ and
$K^{\pm}$ external beams be provided from CERN-LHC, so that their elastic
scattering from protons can be studied at higher energies. Our phenomenology
for $p p$ and $\bar p p$ elastic scattering is updated by including new data
and we give predictions for future experiments at BNL-RHIC and CERN-LHC.
