Duality and Riemannian cubics

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    Abstract

    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) over dot( t) = (beta(0) + t beta(1)) x( t), where beta(0), beta(1) are skew-symmetric 3 x 3 matrices, and x : R --> SO( 3). This is done by showing that the dual of beta(0) + t beta(1) is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics.
    Original languageEnglish
    Pages (from-to)195-209
    JournalAdvances in Computational Mathematics
    Volume25
    Issue number1-3
    DOIs
    Publication statusPublished - 2006

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