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Smooth interpolation on ellipsoids via rolling motions

Fátima Silva Leite, Krzysztof Krakowski
We present an algorithm to generate a smooth curve interpolating a set of data on an n-dimensional ellipsoid. This is inspired by an algorithm based on a rolling and wrapping technique, described in (Huper and Silva Leite, 2007) for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, that algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a geodesic curve. This allows to project data from the ellipsoid to a space where interpolation problems can be easily
solved. The major obstacle to implement the rolling part of that algorithm is due to the fact that explicit forms for geodesics on the ellipsoid with respect to the Euclidean metric are not known. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space on R^{n+1} equipped with an appropriate Riemannian metric, so that geodesics are given in explicit form and the kinematics of the rolling motion are easy to solve. By doing so, we can rewrite the algorithm to generate a smooth interpolating curve on the ellipsoid which is given in closed form.
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