  We give an explicit parametrization of the Hilbert schemes of rational curves
C in P^n having a given splitting type of the restricted tangent bundle from
P^n to C. The adopted technique uses the description of such curves as
projections of a rational normal curve from a suitable linear vertex and a
classification of those vertices that correspond to the required splitting type
of the restricted tangent bundle. This classification involves the study of a
suitable PGL(2) action on the relevant Grassmannian variety.
