In computer graphics the motion of the jaw is commonly modelled by updown and left-right rotation around a fixed pivot plus a forward backward translation, yielding a three dimensional rig that is highly suited for intuitive artistic control. The anatomical motion of the jaw is, however, much more complex since the joints that connect the jaw to the skull exhibit both rotational and translational components. In reality the jaw does not move in a three dimensional subspace but on a constrained manifold in six dimensions. We analyze this manifold in the context of computer animation and show how the manifold can be parameterized with three degrees of freedom, providing a novel jaw rig that preserves the intuitive control while providing more accurate jaw positioning. The chosen parameterization furthermore places anatomically correct limits on the motion, preventing the rig from entering physiologically infeasible poses. Our new jaw rig is empirically designed from accurate capture data, and we provide a simple method to retarget the rig to new characters, both human and fantasy.