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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 8

### Approximating near-geodesic natural cubic splines

Pages: 1409 – 1425

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n8.a2

#### Author

#### Abstract

A method is given for calculating approximations to natural Riemannian cubic splines in symmetric spaces with computational effort comparable to what is needed for the classical case of a natural cubic spline in Euclidean space. Interpolation of $n + 1$ points in the unit sphere $S^m$ requires the solution of a sparse linear system of $4mn$ linear equations. For $n + 1$ points in bi-invariant $SO(p)$ we have a sparse linear system of $2np(p - 1)$ equations. Examples are given for the Euclidean sphere $S^2$ and for bi-invariant $SO(3)$ showing significant improvements over standard chart-based interpolants.

#### Keywords

Lie group, Riemannian manifold, trajectory planning, mechanics, rigid body

#### 2010 Mathematics Subject Classification

34E05, 49K99, 53A99, 70E17, 70E18