Any (arc-length parametrized) space curve is uniquely determined (up to rigid motion) by its curvature  and its torsion. For instance we know that a necessary and sufficient condition for a space curve to lie on a sphere is $R^2+(R')^2T^2=const$, where $R=1/\kappa$, $T=1/\tau$, and $R'$ is the derivative of $R$ relative to $s$. I want to know if there is a necessary and sufficient condition for a space curve to lie on a ellipsoid (in terms of its curvature and torsion).