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Laguerre–Forsyth invariant

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In projective geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom analyzed the invariant in an influential book on ordinary differential equations.

Suppose that is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by then associated to p is the third-order ordinary differential equation

Generically, this equation can be put into the form

where are rational functions of the components of p and its derivatives. After a change of variables of the form , this equation can be further reduced to an equation without first or second derivative terms

The invariant is the Laguerre–Forsyth invariant.

A key property of P is that the cubic differential P(dt)3 is invariant under the automorphism group of the projective line. More precisely, it is invariant under , , and .

The invariant P vanishes identically if (and only if) the curve is a conic section. Points where P vanishes are called the sextactic points of the curve. It is a theorem of Herglotz and Radon that every closed strictly convex curve has at least six sextactic points. This result has been extended to a variety of optimal minima for simple closed (but not necessarily convex) curves by Thorbergsson & Umehara (2002), depending on the curve's homotopy class in the projective plane.

References

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  • Sasaki, Shigeo (1999), Projective differential geometry and linear homogeneous differential equations
  • Thorbergsson, G; Umehara, M (2002), "Sextactic points on a simple closed curve", Nagoya Mathematical Journal, 167 (4): 55–94