Plane quartics: the universal matrix of bitangents

Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special $7$-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. \cite{lh}) proved that a smooth plane quartic can be explicitly reconstructed from its $28$ bitangents; this result improved Aronhold's method of recovering the curve. In a 2011 paper \cite{PSV} Plaumann, Sturmfels and Vinzant introduced an $8 \times 8$ symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse's result for which every smooth quartic curve has exactly $36$ equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e. the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve.