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There has been increasing interest in statistical analysis of data lying inmanifolds. This paper generalizes a smoothing spline fitting method toRiemannian manifold data based on the technique of unrolling and unwrappingoriginally proposed in Jupp and Kent (1987) for spherical data. In particularwe develop such a fitting procedure for shapes of configurations in general$m$-dimensional Euclidean space, extending our previous work for twodimensional shapes. We show that parallel transport along a geodesic on Kendallshape space is linked to the solution of a homogeneous first-order differentialequation, some of whose coefficients are implicitly defined functions. Thisfinding enables us to approximate the procedure of unrolling and unwrapping bysimultaneously solving such equations numerically, and so to find numericalsolutions for smoothing splines fitted to higher dimensional shape data. Thisfitting method is applied to the analysis of some dynamic 3D peptide data.

Smoothing splines on Riemannian manifolds, with applications to 3D shape space