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 $R²+(R')²T²=const$, where $R=1/κ$, $T=1/τ$, 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).