On the reconstruction problem for Pascal lines

TL;DR

This paper demonstrates that a sextuple of points on a conic can be explicitly reconstructed from four specific Pascal lines using transvectant identities and graphical calculus for binary forms.

Contribution

It provides explicit reconstruction formulae for sextuples on a conic from four Pascals, utilizing transvectant identities and graphical calculus.

Findings

Reconstruction of sextuples from four Pascals is possible.

Explicit formulae are derived using transvectant identities.

Graphical calculus aids in proving the identities.

Abstract

Given a sextuple of distinct points $A, B, C, D, E, F$ on a conic, arranged into an array $\left[\begin{array}{ccc} A & B & C F & E & D \end{array}\right]$, Pascal's theorem says that the points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called the Pascal of the array, and one gets altogether sixty such lines by permuting the points. In this paper we prove that the initial sextuple can be explicitly reconstructed from four specifically chosen Pascals. The reconstruction formulae are encoded by some transvectant identities which are proved using the graphical calculus for binary forms.