A New Class of Curves of Rational B-Spline Type

A new class of rational parametrization has been developed and it was used to generate a new family of rational k functions B-splines which depends on an index α ∈ ]−∞ , 0[ ∪ ]1 , +∞[. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when α → ∞. The associated B-splines curves verify the traditional properties particularly that of a convex hull and we see a certain “conjugated symmetry” related to α. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.

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.

Parametric Hull Design with Rational Bézier Curves

In this paper, a tool able to support the sailing yacht designer during the early stage of the design process has been developed. Quadratic and 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 water line, 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) have been conceived aimed to avoid unreasonable shapes. These curves can be imported in any commercial CAD software and used as a frame to fit with a surface. 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 sailboats with different characteristics have been replicated. The rebuilt version of the hulls is very close to the original ones both in terms of shape and dimensionless coefficients.

Neural-Network-Based Curve Fitting Using Totally Positive Rational Bases

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational Bézier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation.

A New Systematic Series of Foil Sections with Parallel Sides

Parallel-sided foil sections are used for centerboards and rudders in sailing dinghy classes and also for struts placed in a fluid flow. The objective of this work is to create a systematic series of parallel-sided sections to be used under different conditions, with an emphasis on the sailing dinghies 470, 420 and Optimist. The loss, and surprisingly the gain, in performance relative to 4-digit NACA sections are also investigated. A 2D Reynolds-averaged Navier–Stokes solver is used with the k-ω SST turbulence model and the gamma transition criterion. A verification study is carried out based on four grids of systematically varied density, and results compared with experimental data on a NACA 64-006 section. The parallel-sided sections are modeled with rational Bézier curves whose geometrical parameters permit to link the shape of the profile to physical variables, which are systematically varied. Three Reynolds numbers and two angles of attack are investigated. Systematic plots show the influence of the trailing edge angle and nose radius for the different section families, and the optimum combination is presented in a table. Physical explanations of the trends, and of the exceptions, are given in the paper, using flow visualizations as well as pressure and friction plots.