  The Bakamjian-Thomas relativistic quark model provides a Poincar\'e
representation of bound states with a fixed number of constituents and, in the
heavy quark limit, form factors of currents satisfy covariance and Isgur-Wise
scaling. We compute the Light Cone Distribution Amplitudes of $B$ mesons
$\phi_{\pm}^B(\omega)$ as well as the Shape Function $S(\omega)$, that enters
in the decay $B \to X_s \gamma$, that are also covariant in this class of
models. The LCDA and the SF are related through the quark model wave function.
The former satisfy, in the limit of vanishing constituent light quark mass, the
integral relation given by QCD in the valence sector of Fock space. Using a
gaussian wave function, the obtained $S(\omega)$ is identical to the so-called
Roman Shape Function. From the parameters for the latter that fit the $B \to
X_s\gamma$ spectrum we predict the behaviour of $\phi_{\pm}^B(\omega)$. We
discuss the important role played by the constituent light quark mass. In
particular, although $\phi_-^B(0) \not= 0$ for vanishing light quark mass, a
non-vanishing mass implies the unfamiliar result $\phi_-^B (0) = 0$. Moreover,
we incorporate the short distance behaviour of QCD to $\phi_+^B (\omega)$,
which has sizeable effects at large $\omega$. We obtain the values for the
parameters $\bar{\Lambda} \cong 0.35$ GeV and $\lambda_B^{-1} \cong 1.43$
GeV$^{-1}$. We compare with other theoretical approaches and illustrate the
great variety of models found in the literature for the functions $\phi_{\pm}^B
(\omega)$; hence the necessity of imposing further constraints as in the
present paper. We briefly review also the different phenomena that are
sensitive to the LCDA.
