  The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian
immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian
submanifold of $Q^n$ is locally the image under the Gauss map of several
hypersurfaces of $S^{n+1}(1)$. In this paper, we give explicit constructions
for these correspondences and we prove a relation between the principal
curvatures of a hypersurface of $S^{n+1}(1)$ and the local angle functions of
the corresponding Lagrangian submanifold of $Q^n$. The existence of such a
relation is remarkable since the definition of the angle functions depends on
the choice of an almost product structure on $Q^n$ and since several
hypersurfaces of $S^{n+1}(1)$, with different principal curvatures, correspond
to the same Lagrangian submanifold of $Q^n$.
