A Dirichlet Process Mixture Model for Spherical Data.

Since the geodesic between any two points is the shortest path on the manifold between the two of them, we know that p has to lie on the geodesic. On the unit sphere we can describe the location on the geodesic as a rotation about the axis defined by the cross product of the two vectors 〈x〉a and 〈x〉b by an angle θa, which we define such that the location of 〈x〉a on the geodesic has θa = 0. This implies that the location of 〈x〉b on the geodesic has angle θb = arccos(〈x〉a 〈x〉b). With this intuition we can reformulate the optimization problem in terms of angles on the geodesic as: