Rigid Body Interpolations and Riemannian Cubics

Lyle Noakes, University of Western Australia, Mathematics

Riemannian cubics are curves in Riemannian manifolds satisfying a particular variational condition, namely they are critical points of the integrated norm-squared covariant acceleration. In Euclidean space Riemannian cubics are polynomial curves of degree at most 3. Riemannian cubics in the rotation group $SO(3)$ are of interest for interpolation of rigid body motion. We review some mathematical background, and describe new results on asymptotic properties, solutions by quadrature from Lie quadratics, and boundary value problems for Riemannian cubics.