On the geometry of Riemannian cubic polynomials

Camarinha, Margarida; Leite, F. Silva; Crouch, P. E.

doi:10.1016/s0926-2245(01)00054-7

Cited by 67 publications

(57 citation statements)

References 9 publications

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“…Interpolation problems on manifolds have been studied by several authors, starting with the pioneer work of Noakes, Heinzinger and Paden in [18]. Following this, other authors further developed the theory of geometric splines on manifolds using a variational approach (see, for instance, [2,4] and [5]). A more general variational problem, that of fitting a curve to data points on a Riemannian manifold, was presented and studied in [17].…”

Section: Introductionmentioning

confidence: 99%

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

Krakowski¹,

Leite²

2014

Kybernetika

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Section: Introductionmentioning

confidence: 99%

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

Krakowski¹,

Leite²

2014

Kybernetika

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“…For comparison, in the Riemannian case, a variational formulation placing a cost on covariant acceleration [28,29] leads to cubic splines…”

Section: ξ(∂ T H U T (U T E D ))mentioning

confidence: 99%

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

Sommer

2016

Entropy

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Abstract:We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton-Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups.

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“…Under this assumption, the resulting trajectories are geometric cubic polynomials on the configuration space of the vehicle. These curves, which are generalizations to Riemannian manifolds of the classical and well established cubic polynomials on Euclidean spaces, have been first introduced by Noakes et al in [7] and further developed, for instance, in [1] and [2]. These optimization problems are formulated via a variational approach and the corresponding Euler-Lagrange equations have been derived in the general context of manifolds.…”

Section: Introductionmentioning

confidence: 99%