In this talk, we present the family of generalized Hessian curves.

The family of generalized Hessian curves covers more isomorphism classes of elliptic curves than Hessian curves.

We provide efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen curve parameters.

We also also present extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require $9\M+3\s$, where $\M$ is the cost of a field multiplication and $\s$ is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only $5\M+4\s$ when the curve is chosen with small parameters.