  We prove that the Hessian transformation of a plane projective cubic
corresponds to a $3$-endomorphism of a model elliptic curve. By exploiting this
result, we investigate a family of functional graphs -- called Hessian graphs
-- defined by the Hessian transformation. We show that, over arbitrary fields
of characteristics different from $2$ and $3$, the Hessian graphs inherit
distinctive features from the arithmetic of the model curve. We then specialize
our analysis to the finite-field case, proving several regularities of Hessian
graphs.
