Intrinsic curvature of an elastic filament (${\kappa }_0(s)$)

The intrinsic signed curvature is the curvature of a stress-free filament, measured along material arc length. An elastic filament with actual curvature $\kappa$ stores bending energy $\frac{B}{2}\int (\kappa -{\kappa }_0{)}^{2}ds$, with filament bending modulus $B$. In a planar small-slope Monge representation, $\kappa \simeq {\zeta }^{\prime \prime }$ and the leading energy is quadratic in ${\zeta }^{\prime \prime }-{\kappa }_0$.