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On the Reconstruction Problem for Pascal Lines

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Abstract

Given a sextuple of distinct points A, B, C, D, E, F 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.

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Notes

  1. One can find a proof in virtually any book on elementary projective geometry, e.g., Pedoe [21, Ch. IX] or Seidenberg [25, Ch. 6]. It is doubtful whether Pascal himself had a proof.

  2. If one tries to draw a diagram of the sextuple together with all 60 of its Pascals, a dense and incomprehensible profusion of ink is the usual outcome. The curious reader is referred to http://mathworld.wolfram.com/PascalLines.html.

  3. The conic itself is fixed throughout, and as such assumed to be known.

  4. It has a close affinity to the classical symbolic calculus as practiced by the German school of invariant theorists in the nineteenth century (cf. [7, 11, 17]). Section 2 of [1] explains the precise correspondence between these two formalisms.

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA, 22904-4137, USA

    Abdelmalek Abdesselam

  2. Department of Mathematics, Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Jaydeep Chipalkatti

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  1. Abdelmalek Abdesselam
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  2. Jaydeep Chipalkatti
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Correspondence to Jaydeep Chipalkatti.

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Abdesselam, A., Chipalkatti, J. On the Reconstruction Problem for Pascal Lines. Discrete Comput Geom 60, 381–405 (2018). https://doi.org/10.1007/s00454-018-9981-4

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  • DOI: https://doi.org/10.1007/s00454-018-9981-4

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