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- Milos Zefran, Vijay Kumar, Christopher Croke
- IEEE Trans. Robotics and Automation
- 1998

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)

- Milos Zefran, Vijay Kumar, Christopher Croke
- I. J. Robotics Res.
- 1999

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)

- C Croke, V Schroeder, +6 authors Viktor Schroeder
- 2010

The fundamental group of compact manifolds without conjugate points.

We present a theorem of Lancret for general helices in a 3-dimensional real-space-form which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3-sphere: the problem of solving natural equations and the closed curve problem.

- VICTOR BANGERT, CHRISTOPHER CROKE, SERGEI V. IVANOV
- 2004

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)

- VICTOR BANGERT, CHRISTOPHER CROKE, SERGEI V. IVANOV
- 2004

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 construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus… (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)