the invariable plane defined by its dot product with the conserved vector . The normal vector to the invariable plane is aligned with . The path traced out by the angular velocity vector on the invariable plane is called the herpolhode (coined from Greek roots for "serpentine pole path"). The herpolhode is generally an open curve, which means that the rotation does not perfectly repeat, but the polhode is a closed curve (see below). Tangency condition and construction These two constraints operate in different reference frames; the ellipsoidal constraint holds in the (rotating) principal axis frame, whereas the invariable plane constant operates in absolute space. To relate these constraints, we note that the gradient vector of the kinetic energy with respect to angular velocity vector equals the angular momentum vector Hence, the normal vector to the kinetic-energy ellipsoid at is proportional to , which is also true of the invariable plane. Since their normal vectors point in the same direction, these two surfaces will intersect tangentially. Taken together, these results show that, in an absolute reference frame, the instantaneous angular velocity vector is the point of intersection between a fixed invariable plane and a kinetic-energy ellipsoid that is tangent to it and rolls around on it without slipping. This is Poinsot's construction. Derivation of the polhodes in the body frame In the principal axis frame (which is rotating in absolute space), the angular momentum vector is not conserved even in the absence of applied torques, but varies as described by Euler's equations. However, in the absence of applied torques, the magnitude of the angular momentum and the kinetic energy are both conserved where the are the components of the angular momentum vector along the principal axes, and the are the principal moments of inertia. These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector . The kinetic energy constrains to lie on an ellipsoid, whereas the angular momentum constraint constrains to lie on a sphere. These two surfaces intersect in two curves shaped like the edge of a taco that define the possible solutions for . This shows that , and the polhode, stay on a closed loop, in the object's moving frame of reference. The orientation of the body in space thus has two degrees of freedom. Firstly, some point on the "taco edge" has to align with which is a constant vector in absolute