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The Fisher distribution is the analogue on the sphere of the isotropic bivariate normal distribution in the plane. The purpose of this paper is to propose and analyse a spherical analogue of the general bivariate normal distribution. Estimation, hypothesis testing and confidence regions are also discussed.with respect to sin e de dfjJ. For plotting purposes Lambert's equal area projection, Z2 = Pcos fjJ, Z3 = Psin fjJ, (1.5) where p = 2 sin (0/2), 0~p~2, is useful because dz1dZ 2 = sin 0 dO dfjJ (see, for example, Mardia, 1972, p. 215). Note that the Lambert projection maps 0 3 onto the disc {(ZI,Z2): (zI +z~)t~2}.

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A complex version of the Bingham distribution is defined on the unit complex sphere in c'. Various statistical properties, including asymptotic normality under high concentration, are derived. Symmetries in the distribution make it a natural tool for the analysis of the shape of landmark data in two dimensions. Strengths and weaknesses of this approach are investigated. Links and contrasts with Bookstein co-ordinates are discussed. A hybrid approach to principal component analysis based on complex and real co-ordinates is suggested.

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Given a set of data points on the sphere at known times, one often wishes to fit a smooth path to the data. In this paper we propose a unified approach to deal with such problems. Our method can be described as "unwrapping" the data onto the plane, where standard curve fitting techniques can then be applied. As an important example of our approach, we define and fit "spherical spline functions" .

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