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Louis Poinsot has shown in 1854 that the motion of a rigid body, with one ofits points fixed, can be described as the rolling without slipping of one cone,the 'body cone', along another, the 'space cone', with their common vertex atthe fixed point. This description has been further refined by the second authorin 1996, relating the geodesic curvatures of the spherical curves formed byintersecting the cones with the unit sphere in Euclidean $\mathbb{R}^3$, thusenabling a reconstruction of the motion of the body from knowledge of the spacecone together with the (time dependent) magnitude of the angular velocityvector. In this article we show that a similar description exists for a timedependent family of unimodular $ 2 \times 2 $ matrices in terms of rollingcones in 3-dimensional Minkowski space $\mathbb{R}^{2,1}$ and the associated'pseudo spherical' curves, in either the hyperbolic plane $H^2$ or itsLorentzian analog $H^{1,1}$. In particular, this yields an apparently newgeometric interpretation of Schr\"odinger's (or Hill's) equation $ \ddot x +q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolicplane.

Hill's equation, tire tracks and rolling cones