Rough cut tool path planning for B-spline surfaces using convex hull boxes

The objective of this paper is to present a non-uniform layered rough cut plan for B-spline surfaces using convex hull boxes. The tool path plan generated by this method will rapidly remove most redundant material from stock material without overcutting. First, a B-spline surface is decomposed into piecewise Bezier surfaces, of which the convex hull boxes form an approximate model for rough cutting. Then, according to the top planes of those convex hull boxes, the stock material is divided into layers of different thickness. The cavity contour for each layer is obtained using a simplified union Boolean operation on convex hull boxes. Finally, from the top down, each layer is processed like a 2D pocket die cavity. The algorithm is implemented on a personal computer. It is shown that the rough cut plan is very efficient since no computation for solving nonlinear equations is needed, and no overcutting occurs since B-spline surfaces are protected by the convex hull property of Bezier surfaces

Reconstruction of interfaces from the elastic farfield measurements using CGO solutions

In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a computable dominant part and an $H^\alpha$-decaying remainder term with $\alpha <3$, where $H^{\alpha}$ is the classical Sobolev space. Second, based on these properties, we estimate the convex hull as well as non convex parts of the interface using the farfields of only one of the two reflected body waves (pressure waves or shear waves) as measurements. The results are given for both the impenetrable obstacles, with traction boundary conditions, and the penetrable obstacles. In the analysis, we require the surfaces of the obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lam\'e coefficients to be measurable and bounded with the usual jump conditions across the interface.Comment: 32 page

Convex hull property and exclosure theorems for H-minimal hypersurfaces in carnot groups

In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition

Semidefinite representation of convex hulls of rational varieties

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension

On the convex hull of a space curve

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing their defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change