SUMMARY This paper presents two families of curves for interpolating and smoothing sequences of points on the surface of the unit sphere. One family is a natural analogue of the usual splines for curve-fitting in the plane; the other is slightly less optimal but more convenient from a computational viewpoint. Data which may be regarded as time-ordered sequences of points on the surface of the sphere arise in a variety of situations in the Earth Sciences. An interesting example of such data is a set of virtual geomagnetic pole positions calculated from rock specimens of different ages, for a single continent. The chronological sequence of these pole positions is known as the Apparent Polar Wander Path. Comparison of such paths for two continents, over the same time period, is some- times used to decide whether the two continents have moved relative to each other over this period. Various authors have discussed the problem of fitting curves piecewise to sequences
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