Search

Quartic C-Bézier curves possess similar properties with the traditional Bézier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter α decreases. In this paper, by adjusting the shape parameter α on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C-Bézier curve with G1 continuity of quartic C-Bézier curve segments, we develop a fairing and G1 continuity algorithm for any given stitching coefficients λ k( k = 1,2, ..., n − 1). The shape parameters α i( i=1, 2, ..., n) can be adjusted by the value of control points. The curvature of the resulting quartic C-Bézier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer-aided design/computer-aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the estimation of the character sums to study the 2k-th power mean of L-functions with the weight of a character sums, and give an interesting asymptotic formula for it. [ABSTRACT FROM AUTHOR]

Abstract: A new numerical method for heat conduction equations has been developed by utilizing the collocation method and a new ridge basis function interpolation. The method is applied to 1/2-dimensional equations to examine its appropriateness, and influence factors of accuracy for numerical solutions are analyzed. [Copyright &y& Elsevier]

SG-Bézier curves have become a useful tool for shape design and geometric representation in computer aided design (CAD), owed to their good geometric properties, e.g., symmetry and convex hull property. Aiming at the problem of approximate degree reduction of SG-Bézier curves, a method is proposed to reduce the n-th SG-Bézier curves to m-th (m < n) SG-Bézier curves. Starting from the idea of grey wolf optimizer (GWO) and combining the geometric properties of SG-Bézier curves, this method converts the problem of multi-degree reduction of SG-Bézier curves into solving an optimization problem. By choosing the fitness function, the approximate multi-degree reduction of SG-Bézier curves with adjustable shape parameters is realized under unrestricted and corner interpolation constraints. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect, but is also easy to implement and has high accuracy. [ABSTRACT FROM AUTHOR]

The S-λ Curves have become an important research subject in computer aided geometric design (CAGD), which owes to its good geometric properties (such as affine invariance, symmetry, and locality). This paper presents a new method to approximate an S-λ curve of degree n by using an S-λ curve of degree n-1. We transform this degree reduction problem into the function optimization problem first, and then using a new genetic simulated annealing algorithm to determine the global optimal solution of the optimization problem. The method can be used to approximate S-λ curves with fixed or unconstrained endpoints. Examples are given to verify the effectiveness of the presented algorithm; and these numeric examples show that the algorithm is not only easy to implement, but also offers high precision, which makes it valuable in practical applications. [ABSTRACT FROM AUTHOR]

Besides inheriting the properties of classical Bézier curves of degree n, the corresponding λ-Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, the λ-Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification of λ-Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties of λ-Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm, λ-Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape of λ-Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement. [ABSTRACT FROM AUTHOR]