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