A Constructive Approximation of Interpolating Bézier Curves on Riemannian Symmetric Spaces

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We propose a new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces. First, we present our new formulation on the Euclidean space as a result of minimizing the mean square acceleration. This motivates its generalization on some Riemannian symmetric manifolds. Indeed, we generalize the proposed solution to the the special orthogonal group, the manifold of symmetric positive definite matrices, and Riemannian *n*-manifolds with constant negative curvature. By means of this generalization, we are able to prove that the approximates enjoy a number of nice properties: The solution exists and is optimal in many common situations. Several examples are provided together with some applications and graphical representations.