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Quaternions, Haar Measure and the Estimation of a Palaeomagnetic Rotation

Representing a rotation in three dimensions by a unit tensor quaternion and supposing that the errors in the measurements of a number of vector directions follow Fisher's distribution, the maximum likelihood estimator of a rotation is obtained. It is shown that the natural Haar measure for three-dimensional rotations is mapped 1:2 onto the natural Haar measure for random directions in 4-space. The natural Haar measure for rotations in 4-space is also mappable onto the product measure of the Haar measures for two separate random directions in 4-space.