B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure

A discrete methodology for controlling the sign of curvature and torsion for NURBS

This paper develops a discrete methodology for approximating the so-called convex domain of a NURBS curve, namely the domain in the ambient space, where a user-specified control point is free to move so that the curvature and torsion retains its sign along the NURBS parametric domain of definition. The methodology provides a monotonic sequence of convex polyhedra, converging from the interior to the convex domain. If the latter is non-empty, a simple algorithm is proposed, that yields a sequence of polytopes converging uniformly to the restriction of the convex domain to any user-specified bounding box. The algorithm is illustrated for a pair of planar and a spatial Bézier configuration

Optimization of a Centrifugal Compressor Using the Design of Experiment Technique

Centrifugal compressor performance is affected by many parameters, optimization of which can lead to superior designs. Recognizing the most important parameters affecting performance helps to reduce the optimization process cost. Of the compressor components, the impeller plays the most important role in compressor performance, hence the design parameters affecting this component were considered. A turbocharger centrifugal compressor with vaneless diffuser was studied and the parameters investigated included meridional geometry, rotor blade angle distribution and start location of the main blades and splitters. The diffuser shape was captured as part of the meridional geometry. Applying a novel approach to the problem, full factorial analysis was used to investigate the most effective parameters. The Response Surface Method was then implemented to construct the surrogate models and to recognize the best points over a design space created as based on the Box-Behnken methodology. The results highlighted the factors that affected impeller performance the most. Using the Design of Experiment technique, the model which optimized both efficiency and pressure ratio simultaneously delivered a design with 3% and 11% improvement in each respectively in comparison to the initial impeller at the design point. Importantly, this was not at the expense of sacrificing range, of critical concern in compressor design

Parametric Hull Design with Rational Bézier Curves and Estimation of Performances

In this paper, a tool able to support the sailing yacht designer during the early stage of the design process has been developed. Cubic Rational Bézier curves have been selected to describe the main curves defining the hull of a sailing yacht. The adopted approach is based upon the definition of a set of parameters, say the length of waterline, the beam of the waterline, canoe body draft and some dimensionless coefficients according to the traditional way of the yacht designer. Some geometrical constraints imposed on the curves (e.g., continuity, endpoint angles, curvature) have been conceived aimed to avoid unreasonable shapes. These curves can be imported into any commercial Computer Aided Design (CAD) software and used as a frame to fit with a surface. The resistance of the hull can be calculated and plotted in order to have a real time estimation of the performances. The algorithm and the related Graphical User Interface (GUI) have been written in Visual Basic for Excel. To test the usability and the precision of the tool, two existing sailboats with different characteristics have been successfully replicated and a new design, taking advantages of both the hulls, has been developed. The new design shows good performances in terms of resistance values in a wide range of Froude numbers with respect to the original hulls

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Parametric Interpolation To Scattered Data [QA281. A995 2008 f rb].

Dua skema interpolasi berparameter yang mengandungi interpolasi global untuk data tersebar am dan interpolasi pengekalan-kepositifan setempat data tersebar positif dibincangkan. Two schemes of parametric interpolation consisting of a global scheme to interpolate general scattered data and a local positivity-preserving scheme to interpolate positive scattered data are described

Cubic B-spline curve approximation by curve unclamping

International audienceA new approach for cubic B-spline curve approximation is presented. The method produces an approximation cubic B-spline curve tangent to a given curve at a set of selected positions, called tangent points, in a piecewise manner starting from a seed segment. A heuristic method is provided to select the tangent points. The first segment of the approximation cubic B-spline curve can be obtained using an inner point interpolation method, least-squares method or geometric Hermite method as a seed segment. The approximation curve is further extended to other tangent points one by one by curve unclamping. New tangent points can also be added, if necessary, by using the concept of the minimum shape deformation angle of an inner point for better approximation. Numerical examples show that the new method is effective in approximating a given curve and is efficient in computation

Accurate and efficient spin integration for particle accelerators

Accurate spin tracking is a valuable tool for understanding spin dynamics in particle accelerators and can help improve the performance of an accelerator. In this paper, we present a detailed discussion of the integrators in the spin tracking code gpuSpinTrack. We have implemented orbital integrators based on drift-kick, bend-kick, and matrix-kick splits. On top of the orbital integrators, we have implemented various integrators for the spin motion. These integrators use quaternions and Romberg quadratures to accelerate both the computation and the convergence of spin rotations. We evaluate their performance and accuracy in quantitative detail for individual elements as well as for the entire RHIC lattice. We exploit the inherently data-parallel nature of spin tracking to accelerate our algorithms on graphics processing units.Comment: 43 pages, 17 figure