For torque-free rigid-body motion, the kinetic-energy ellipsoid fixed in the body rolls without slipping on the invariable plane perpendicular to the conserved angular momentum. Its point of contact is the angular-velocity vector.
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Poinsot's ellipsoid is a geometric representation used in the study of rigid bodies in classical mechanics. It specifically describes the distribution of mass and the inertial properties of a rigid body. The concept is related to the inertia tensor of the body, which characterizes how the mass is distributed relative to its rotational axes. Poinsot's ellipsoid represents the relationships between the principal moments of inertia and the axes of rotation.