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A poncelet theorem in space

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Commentarii Mathematici Helvetici

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References

  1. J. V. Poncelet,Traité des propriétés projectives des figures, Mett-Paris 1822.

  2. C. G. J. Jacobi,Über die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie, Gesammelte Werke, Vol.I (1881), pp. 278–293.

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  3. P. Griffiths,Variations on a theorem of Abel, Inventiones Math. Vol. 35 (1976), ppg. 321–390. This paper also contains a variant of Jacobi's discussion of the classical Poncelet problem.

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  4. We shall allow the polygon to have self-intersections.

  5. This may be verified analytically using the equations ofS andS' given below.

  6. We note thatS ∩T p (S) is a plane conic containing a line, and hence must be two lines throughP as in Fig. 6. These lines are distinct since detQ#0.

  7. SinceS∩(LΝL') contains and therefore is equal to the plane conicL+L', it follows thatLΝL' is the tangent plane toS atL∩L'. In particular,L 0, does not lie inLΝL' nor pass throughL∩L'.

  8. We will comment on the construction in ℝ3 at the end of the paper.

  9. The vertices are where three or more planes meet. In Fig. 8 these are the vertices of the shaded quadrilaterals.

  10. This is the formulaK D =K M ⊗[D] D for the canonical line bundle of a smooth divisorD on a complex manifoldM.

  11. That the intersection is transverse follows from the equations forS andS' given below.

  12. In this connection M. Berger pointed out two papers by Cayley giving explicit conditions for Poncelet polygon to be closed. The references are Philosophical Magazine, vol. VI (1853), 99–102. and Philosophical Trans. Royal Soc. London, vol. CLI (1861), 225–239.

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Authors and Affiliations

  1. Harvard University, Berkeley, USA

    Phillip Griffiths

  2. the Miller Institute for Basic Research In Science, University of California at Berkeley, Berkeley, USA

    Phillip Griffiths

Authors
  1. Phillip Griffiths
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  2. Joe Harris
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Research partially supported by NSF grant GP 38886.

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Griffiths, P., Harris, J. A poncelet theorem in space. Commentarii Mathematici Helvetici 52, 145–160 (1977). https://doi.org/10.1007/BF02567361

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  • DOI: https://doi.org/10.1007/BF02567361

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