Angle between principal axis triples

S U M M A R Y The principal axis angle ξ 0, or Kagan angle, is a measure of the difference between the orientations of two seismic moment tensors. It is the smallest angle needed to rotate the principal axes of one moment tensor to the corresponding principal axes of the other. This paper is a conceptual review of the main features of ξ 0. We give a concise formula for calculating ξ 0, but our main goal is to illustrate the behaviour of ξ 0 geometrically. When the first of two moment tensors is fixed, the angle ξ 0 between them becomes a function on the unit ball. The level surfaces of ξ 0 can then be depicted in the unit ball, and they give insights into ξ 0 that are not obvious from calculations alone. We also include a derivation of the known probability density pξ0 of ξ 0. The density pξ0 (t) is proportional to the area of a certain surface Lw(t). The easily seen variation of Lw(t) with t then explains the rather peculiar shape of pξ0 . Because the curve pξ0 is highly non-uniform, its shape needs to be considered when analysing distributions of empirical ξ 0 values. We recall an example of Willemann which shows that ξ 0 may not always be the most appropriate measure of separation for moment tensor orientations, and we offer an alternative measure.