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Commentarii Mathematici Helvetici

, Volume 52, Issue 1, pp 145–160 | Cite as

A poncelet theorem in space

  • Phillip Griffiths
  • Joe Harris
Article

Keywords

Riemann Surface Tangent Plane Tangent Line Algebraic Curve Symmetric Pair 
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References

  1. 1.
    J. V. Poncelet,Traité des propriétés projectives des figures, Mett-Paris 1822.Google Scholar
  2. 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.Google Scholar
  3. 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.zbMATHCrossRefGoogle Scholar
  4. 4.
    We shall allow the polygon to have self-intersections.Google Scholar
  5. 5.
    This may be verified analytically using the equations ofS andS' given below.Google Scholar
  6. 6.
    We note thatST 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.Google Scholar
  7. 7.
    SinceS∩(LΝL') contains and therefore is equal to the plane conicL+L', it follows thatLΝL' is the tangent plane toS atLL'. In particular,L 0, does not lie inLΝL' nor pass throughLL'.Google Scholar
  8. 8.
    We will comment on the construction in ℝ3 at the end of the paper.Google Scholar
  9. 9.
    The vertices are where three or more planes meet. In Fig. 8 these are the vertices of the shaded quadrilaterals.Google Scholar
  10. 10.
    This is the formulaK D=K M ⊗[D] D for the canonical line bundle of a smooth divisorD on a complex manifoldM.Google Scholar
  11. 11.
    That the intersection is transverse follows from the equations forS andS' given below.Google Scholar
  12. 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.Google Scholar

Copyright information

© Birkhäuser Verlag 1977

Authors and Affiliations

  • Phillip Griffiths
    • 1
    • 2
  • Joe Harris
  1. 1.Harvard UniversityBerkeleyUSA
  2. 2.the Miller Institute for Basic Research In ScienceUniversity of California at BerkeleyBerkeleyUSA

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