The rotation group in plate tectonics and the representation of uncertainties of plate reconstructions

The calculation of the uncertainty in an estimated rotation requires a parametrization of the rotation group; that is, a unique mapping of the rotation group to a point in 3-D Euclidean space, R^3. Numerous parametrizations of a rotation exist, including: (1) the latitude and longitude of the axis of rotation and the angle of rotation; (2) a representation as a Cartesian vector with length equal to the rotation angle and direction parallel to the rotation axis; (3) Euler angles; or (4) unit length quaternions (or, equivalently, Cayley-Klein parameters). The uncertainty in a rotation is determined by the effect of nearby rotations on the rotated data. The uncertainty in a rotation is small, if rotations close to the best fitting rotation degrade the fit of the data by a large amount, and it is large, if only rotations differing by a large amount cause such a degradation. Ideally, we would like to parametrize the rotations in such a way so that their representation as points in R^3 would have the property that the distance between two points in R3 reflects the effects of the corresponding rotations on the fit of the data. It can be shown mathematically that this is impossible, but for rotations of small angle, it can be done to close approximation by using vectors in Cartesian coordinates. Thus, we are led to parametrizing the uncertainty separately from the parametrization of the best fitting rotation. This approach results in simpler, more efficient calculations than if uncertainties are described in terms of rotation parameters (i.e., latitude, longitude, and the angle). We illustrate this with the example of equations for determining the uncertainty in a composite rotation from the uncertainties of its constituents.