  We prove that any planar projective curve of degree $d \ge 4$ and with a
smooth Hessian curve $H_f$ is uniquely determined by $H_f$. Taking into account
that the Hessian curve is the ramification divisor associated with the polar
map, we prove the statement using a geometric description of the graph of such
a map. It turns out that in degree $d\ge 4$ two curves have the same smooth
Hessian if and only if they have the same Jacobian ideal. The latter curves
have been classified by C. Mammana, and they all have a reducible, hence
singular, Hessian curve.
  As a consequence, the Hessian map, associating with a proportionality class
of a ternary form the class of its Hessian determinant, is injective on the
open locus corresponding to curves with smooth Hessian curve. This confirms a
conjecture posed by C. Ciliberto and G. Ottaviani.
