Local coordinates

Abstract: Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … x N ) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

Abstract: We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with (alpha) metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on novel estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient, that hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications.

Abstract: A system for point-by-point measurement of spatial coordinates comprising: one or more opto-electronic cameras (7, 8) arranged for measuring spatial direction to point-shaped light sources; one or more rangefinders (11) for measuring distance and optionally direction to light-reflecting targets (19, 28); a touch tool (18) having a minimum of three point-shaped light giving means (21-25) in known local coordinates relative to a local tool-fixed coordinate system, one or more light-reflecting points or targets (28) for the laser rangefinder, and having a contact point (30) in known position relative to said local coordinate system; and data processor (2) designed to compute the spatial position and orientation of said touch tool (18) relative to said camera (7, 8) and rangefinder (11) on the basis of knowledge of the position of said light giving means (21-25) relative to the tool's contact point (30), the measured directions from the cameras (7, 8) to the individual light giving means (21-25), and measured distance from the laser rangefinder (11) to the light-reflecting point/target (28), so that the position of the tool (18) is referred to said contact point (30).

Abstract: Introduction. Mathematical Preliminaries. Variational Methods. Element Interpolation and Local Coordinates. One-Dimensional Integration. Beam Analysis. Truss Elements and Axis Transformations. Cylindrical Analysis. General Interpolation. Adaptive Analysis. Integration Models. Heat Transfer. Elasticity. Error Measures for Elliptic Problems. Sensitivity Analysis. Special Elements. Transient Problems. Computational Fluid Dynamics. Automatic Mesh Generation. Computational Procedures. MODEL Applications in 1-D. MODEL Applications in 2-D and 3-D. Appendices. Index.