Constructing reparametrization invariant metrics on spaces of plane curves

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space $\text{Imm}(S^1,\mathbb R^2)$ of parametrized plane curves and the quotient space $\text{Imm}(S^1,\mathbb R^2)/\text{Diff}(S^1)$ of unparametrized curves. For the space of open parametrized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parametrized and are non-negative on the space of unparametrized open curves. For the metric, which is induced by the "R-transform", we provide a numerical algorithm that computes geodesics between unparameterised, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests that demonstrate it's efficiency and robustness.Comment: 27 pages, 4 figures. Extended versio

A and B branes from N=2 superspace

We present a manifestly supersymmetric description of A and B branes on Kaehler manifolds using a completely local N=2 superspace formulation of the world-sheet nonlinear sigma-model in the presence of a boundary. In particular, we show that an N=2 superspace description of type A boundaries is possible. This leads to a concrete realization of the still poorly understood coisotropic A branes. We also discuss briefly how the superspace description of a B brane provides an efficient way to compute higher loop beta-functions. In particular, we sketch how one obtains the fourth order derivative correction to the Born-Infeld action by using a beta-function method.Comment: 8 pages, contribution to the proceedings of the Third Workshop of the RTN project 'Constituents, Fundamental Forces and Symmetries of the Universe', Valencia, October 1 - 5, 200

Cooperative Curve Tracking in Two Dimensions Without Explicit Estimation of the Field Gradient

We design a control law for two agents to successfully track a level curve in the plane without explicitly estimating the field gradient. The velocity of each agent is decomposed along two mutually perpendicular directions, and separate control laws are designed along each direction. We prove that the formation center will converge to the neighborhood of the level curve with the desired level value. The algorithm is tested on some test functions used in optimization problems in the presence of noise. Our results indicate that in spite of the control law being simple and gradient-free, we are able to successfully track noisy planar level curves fast and with a high degree of accuracy.Comment: 4th International Conference on Control, Decision, and Information Technologies (CoDIT) 201