Determining Riemannian cubics and cubic splines with given boundary conditions

Riemannian cubics are critical points of the total squared norm of the accelerations of the curves on Riemannian manifolds with given end-points and end-velocities. Riemannian cubic splines are $C^2$ track sums of Riemannian cubics. Determining Riemannian cubics with given boundary data is equivalent to solving fourth-order ordinary differential equations with boundary conditions, while finding Riemannian cubic splines is tantamount to solving boundary and interior value problems, which are both hard in practice. The present paper proposes Riemannian gradient-based methods to tackle the former problem by taking advantage of the variational principle and the shooting method. The core idea is to add a number of junctions and associated velocities between given boundary points and to adjust such junctions and associated velocities according to a variational principle. Based on the obtained Riemannian cubics, Riemannian cubic splines are determined by constructing piecewise Riemannian cubics of class $C^2$ at interior points. The effectiveness of the proposed method is assessed by numerical experiments on a unit sphere.