# Blended smoothing splines on Riemannian manifolds

## TL;DR

This paper introduces a method for fitting smooth curves on Riemannian manifolds by blending Euclidean Bézier curves in tangent spaces, ensuring smoothness and natural cubic spline properties, demonstrated on the sphere S2.

## Contribution

The paper proposes a novel blending approach for smoothing splines on Riemannian manifolds that guarantees smoothness and reduces to classical splines in Euclidean space.

## Key findings

- Method produces C1 smooth curves on manifolds.
- Ensures the curve is a natural cubic smoothing spline in Euclidean space.
- Demonstrated successfully on the sphere S2.

## Abstract

We present a method to compute a fitting curve B to a set of data points d0,...,dm lying on a manifold M. That curve is obtained by blending together Euclidean B\'ezier curves obtained on different tangent spaces. The method guarantees several properties among which B is C1 and is the natural cubic smoothing spline when M is the Euclidean space. We show examples on the sphere S2 as a proof of concept.

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Source: https://tomesphere.com/paper/1812.04420