Euler's polhode is the time dependent angular velocity curve W(t) in the coordinate system
of the rotating object. One chooses the eigen frame (e1,e2,e3) of the tensor of inertia to have a simple relation
between angular velocity W=(W1,W2,W3), angular momentum L=(L1,L2,L3) and the principal moments of inertia D1,D2,D3:

L1 = D1*W1, L2 = D2*W2, L3 = D3*W3.

The polhode is the solution of Eulers ODE: L' = L x W. Once the angular velocity (W1,W2,W3) is known in the body frame (e1, e2, e3), the rotation in observer space is given by (e1, e2, e3)'(t) = (W1*e1 + W2*e2 + W3*e3)(t) x (e1, e2, e3)(t). Euler's ODE guarantees conservation of angular momentum L in this rotation and also conservation of L*L and of kinetic energy E = 0.5L*W. The polhode is therefore the intersection of a sphere and an ellipsoid (in angular momentum coordinates, otherwise two ellipsoids). A more detailed explanation of how W(t) controls the movement of the object is given here:

From angular velocity to rotational motion.

The definition of the tensor of inertia and the derivation of Euler's ODE is explained here: Free rotational motion of rigid bodies.

L1 = D1*W1, L2 = D2*W2, L3 = D3*W3.

The polhode is the solution of Eulers ODE: L' = L x W. Once the angular velocity (W1,W2,W3) is known in the body frame (e1, e2, e3), the rotation in observer space is given by (e1, e2, e3)'(t) = (W1*e1 + W2*e2 + W3*e3)(t) x (e1, e2, e3)(t). Euler's ODE guarantees conservation of angular momentum L in this rotation and also conservation of L*L and of kinetic energy E = 0.5L*W. The polhode is therefore the intersection of a sphere and an ellipsoid (in angular momentum coordinates, otherwise two ellipsoids). A more detailed explanation of how W(t) controls the movement of the object is given here:

From angular velocity to rotational motion.

The definition of the tensor of inertia and the derivation of Euler's ODE is explained here: Free rotational motion of rigid bodies.