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 formula_1 is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by formula_2 then associated to "p" is the third-order ordinary differential equation Generically, this equation can be put into the form where formula_5 are rational functions of the components of  p and its derivatives. After a change of variables of the form formula_6, this equation can be further reduced to an equation without first or second derivative terms The invariant formula_8 is the Laguerre–Forsyth invariant. A key property of is that the cubic differential is invariant under the automorphism group formula_9 of the projective line. More precisely, it is invariant under formula_10, formula_11, and formula_12. The invariant vanishes identically if (and only 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 (but not necessarily convex) curves by , depending on the curve's homotopy class in the projective plane.