Laguerre–Forsyth invariant

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 is that the cubic differential is invariant under the automorphism group of the projective line. More precisely, it is invariant under,, and.
The invariant vanishes identically if the curve is a conic section. Points where 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 curves by, depending on the curve's homotopy class in the projective plane.