  For a closed surface S, the Hitchin component Hit_n(S) is a preferred
component of the character variety consisting of group homomorphisms from the
fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a
parametrization of the Hitchin component that is well-adapted to a maximal
geodesic lamination on the surface. This is a natural extension of Thurston's
parametrization of the Teichmueller space of S by shear coordinates associated
to a maximal geodesic lamination, corresponding to the case n=2. However,
significantly new ideas are needed in this higher dimensional case. The article
concludes with a few applications.
