Fitting Smooth Paths to Rotation Data

In the study of plate tectonics in geophysics, estimates are available of the cumulative motion of continental plates relative to each other between various geological epochs. For a given pair of plates this relative movement may be summarized at each time by a 3× 3 proper rotation matrix describing the orientation of a plate relative to its initial position. The general problem of point and interval rotation estimation at other times is addressed by using a spline method for fitting smooth paths to such sequences of cumulative rotations. The parameterization of 3× 3 rotations as unsigned four-dimensional directions is particularly convenient. The preferred tangent space projection also makes easy the construction of point and interval estimates of the instantaneous angular velocity vector of the relative motion.