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
p:P1\toP2
p(t)=(x1(t),x2(t),x3(t))
\left|\begin{matrix} x&x'&x''&x'''\\ x1&x1'&x1''&x1'''\\ x2&x2'&x2''&x2'''\\ x3&x3'&x3''&x3'''\\ \end{matrix}\right|=0.
x'''+Ax''+Bx'+Cx=0
A,B,C
t\tof(t),x\tog(t)-1x
x'''+Rx=0.
P=(f')2R
A key property of is that the cubic differential is invariant under the automorphism group
PGL(2,R)
t\to | at+b |
ct+d |
dt\to | ad-bc |
(ct+d)2 |
dt
x\toC(ct+d)-2x
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.