On the Reconstruction Problem for Pascal Lines

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Abstract

Given a sextuple of distinct points ABCDEF on a conic, arranged into an array \(\big [ \begin{array}{ccc} A &{} B &{} C\\ F &{} E &{} D \end{array} \big ],\) 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 60 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.

Keywords

Pascal lines Transvectants Invariant theory of binary forms 

Mathematics Subject Classification

14N05 22E70 51N35 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsMachray Hall, University of ManitobaWinnipegCanada

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