Local barycentric coordinates

Barycentric coordinates yield a powerful and yet simple paradigm to interpolate data values on polyhedral domains. They represent interior points of the domain as an affine combination of a set of control points, defining an interpolation scheme for any function defined on a set of control points. Numerous barycentric coordinate schemes have been proposed satisfying a large variety of properties. However, they typically define interpolation as a combination of all control points. Thus a local change in the value at a single control point will create a global change by propagation into the whole domain. In this context, we present a family of local barycentric coordinates (LBC), which select for each interior point a small set of control points and satisfy common requirements on barycentric coordinates, such as linearity, non-negativity, and smoothness. LBC are achieved through a convex optimization based on total variation, and provide a compact representation that reduces memory footprint and allows for fast deformations. Our experiments show that LBC provide more local and finer control on shape deformation than previous approaches, and lead to more intuitive deformation results

A reduced Hsieh–Clough–Tocher element with splitting based on an arbitrary interior point

AbstractWe present formulas for a reduced Hsieh–Clough–Tocher (rHCT) element with splitting based on an arbitrary interior point. These formulas use local barycentric coordinates in each of the subtriangles and are not significantly more complicated than formulas for an rHCT element with splitting based on the centroid

Relativistic Reference Frames for Astrometry and Navigation in the Solar System

Astrophysical space missions deliver invaluable information about our universe, stellar dynamics of our galaxy, and motion of celestial bodies in the solar system. Astrometric space missions SIM and Gaia will determine distances to stars and cosmological objects as well as their physical characteristics and positions on the celestial sphere with microarcsecond precision. These and other space missions dedicated to exploration of the solar system are invaluable for experimental testing of general relativity. Permanently growing accuracy of space and ground-based astronomical observations require corresponding development of relativistic theory of reference frames, motion of celestial bodies, and propagation of light/radio signals from a source of light/radio to observer. Such theory must be based on Einstein's general relativity and account for various relativistic effects both in the solar system and outside of its boundary. We describe a hierarchy of the relativistic frames adopted by the International Astronomical Union in 2000, and outline directions for its theoretical and practical extentions by matching the IAU 2000 reference frames in the solar system to the cosmological Friedman-Robertson-Walker reference frame and to the frames used in the parametrized post-Newtonian formalism.Comment: 16 pages, bugs in equations removed, minor changes in text, to appear in Proc. of the ASTROCON 2006 meeting (Princeton University, Princeton, NJ, USA) http://www.andrew.cmu.edu/user/jarrieta/blog/astrocon2006.shtm