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).

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).

Since any (arc-length parametrized) space curve is uniquely determined (up to rigid motions) by its curvature  and its torsion. And we knew that a necessary and sufficient condition for a space curve lies 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(a equation of its curvature and torsion)

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).

Since any (arc-length parametrized) space curve is uniquely determined (up to rigid motions) by its curvature  and its torsion  . And we knew that a necessary and sufficient condition for a space curve lies on a sphere is R²+(R')²T²=const , where R=1/κ,T=1/τ,and R' is the derivative of R ralative to s. I want to know if there is a necessary and sufficient condition for a space curve to lie on a ellipsoid(a equation of its curvature and torsion)

Since any (arc-length parametrized) space curve is uniquely determined (up to rigid motions) by its curvature  and its torsion. And we knew that a necessary and sufficient condition for a space curve lies 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(a equation of its curvature and torsion)