Variational time discretization of Riemannian splines

TL;DR

This paper extends cubic splines to Riemannian manifolds by defining a spline energy that combines path energy and covariant derivative, and develops a variational discretization with proven convergence.

Contribution

It introduces a variational time discretization for Riemannian splines, establishing existence and convergence of discrete spline paths to continuous ones.

Findings

Existence of continuous and discrete Riemannian splines proven.

Discrete spline paths converge to continuous splines via $ ext{Gamma}$-convergence.

Applications demonstrated on finite-dimensional, shell, and shape manifolds.

Abstract

We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy - a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity - under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the $\Gamma$-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.