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 quadraticV 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 \(\dot{x}(t)=(\beta_{0}+t\beta_{1})x(t)\) , 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.
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Communicated by Y. Xu
To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday
Mathematics subject classifications (2000)
53A17, 53B20, 65D18, 68U05, 70E60.
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Noakes, L. Duality and Riemannian cubics. Adv Comput Math 25, 195–209 (2006). https://doi.org/10.1007/s10444-004-7621-4
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DOI: https://doi.org/10.1007/s10444-004-7621-4