# A Numerical Algorithm for C2-splines on Symmetric Spaces

## TL;DR

This paper introduces an efficient iterative numerical algorithm for constructing C2-splines on Riemannian symmetric spaces, addressing a key challenge in manifold interpolation with applications in computer vision and quantum control.

## Contribution

It provides the first convergent iterative method for C2-splines on symmetric spaces, surpassing existing approaches in efficiency and parallelizability.

## Key findings

- Algorithm converges linearly.
- Applicable to spheres, complex projective spaces, and Grassmannians.
- Outperforms Riemannian cubic-based methods in computational efficiency.

## Abstract

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, require the solution of a coupled set of non-linear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De~Casteljau's algorithm, which leads to generalized B\'ezier curves. To construct C2-splines from such curves is a complicated non-linear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel, thus suitable for multi-core implementation. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.