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 x(t) = (0 + t1)x(t), where 0, 1 are skew-symmetric 3

• Type: Journal
• Year: 2006
• Where: ADCM
• Authors: Lyle Noakes
• Keywords: ADCM 2006, Lie Quadratics, Linking Equation, Riemannian Cubic