  The Hessian map is the rational map that sends a homogeneous polynomial to
the determinant of its Hessian matrix. We prove that the Hessian map is
birational on its image for ternary forms of degree $d\ge 4$, $d\neq 5$, by
considering the action of the orthogonal group. In a previous paper we proved
the analogous result for binary forms, with more geometric techniques.
