Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physic

Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space

An invariant characterization of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space is given in terms of invariants and covariants of a real binary quartic canonically associated to the characteristic conformal Killing tensor which defines the webs.Comment: 25 pages, recently submitted to the Journal of Mathematical Physic

Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi