  Densities and transition densities are computed in an equilateral triangular
alpha-cluster model for $^{12}$C, in which each $\alpha$ particle is taken as a
gaussian density distribution. The ground-state, the symmetric vibration (Hoyle
state) and the asymmetric bend vibration are analyzed in a molecular approach
and dissected into their components in a series of harmonic functions,
revealing their intrinsic structures. The transition densities in the
laboratory frame are then used to construct form-factors and to compute DWBA
inelastic cross-sections for the $^{12}$C$(\alpha, \alpha')$ reaction. The
comparison with experimental data indicates that the simple geometrical model
with rotations and vibrations gives a reliable description of reactions where
$\alpha$-cluster degrees of freedom are involved.
