Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1, 2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form ẋ(t) = (β0 + tβ1)x(t), where β0, β1 are skew-symmetric 3 × 3 matrices, and x : R → SO(3). This is done by showing that the dual of β0 + tβ1 is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics.