Christopher Croke

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On the generation of smooth three-dimensional rigid body motions. Abstract This paper addresses the problem of generating smooth trajectories between an initial and final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that(More)
The set of rigid body motions forms the Lie group SE(3), the special Euclidean group in three dimensions. In this paper we investigate Riemannian metrics and aane connections on SE(3) that are suited for kinematic analysis and robot trajectory planning. In the rst part of the paper, we study metrics whose geodesics are screw motions. We prove that no(More)
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, g), depending on a pair of parameters , n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with(More)
We prove the filling area conjecture in the hyperellip-tic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit(More)
We prove that the flat product metric on D n × S 1 is scattering rigid where D n is the unit ball in R n and n ≥ 2. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map S : U + ∂M → U − ∂M from unit vectors V at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes V(More)
Let Ω be an (n + 1)-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary M = ∂Ω. Assume that the principal curvatures of M are bounded from below by a positive constant c. In this paper, we prove that the first nonzero eigenvalue λ 1 of the Laplacian of M acting on functions on M satisfies λ 1 ≥ nc 2 with equality(More)
It is proved that any mapping of an n-dimensional affine space over a division ring D onto itself which maps every line into a line is semi-affine, if n ∈ {2, 3,. .. } and D = Z 2. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary(More)