Numerical integration using Bezier splines

As the B-spline method was developed for solving higher order differential equations, we present a brief survey to construct a higher degree B-spline. The new technique has been given in this field, accordingly a numerical illustration used to solve boundary value problems by employ quintic B-spline function. An example has been given for calculating maximum absolute error through n nodes.

Journal of Zankoy Sulaimani - Part A, 2016

A numerical algorithm is constructed to develop numerical solution to the spline function based belonging to the C 6-class. The presented method showed that the approximate solution for boundary value problems obtain by the numerical algorithm which are applied sixtic spline function is effective. Convergence analysis of the proposed method and error estimates are obtained. Numerical results illustrate by two examples are given the practical usefulness and efficiency of the algorithm.

International Journal of …

In this paper, a new quintic spline method developed for computing approximate solution of differential equations. It is shown that the present method is of the order three and four derivatives and gives approximations which are better. The numerical result obtained by the present method has been compared with the exact solution using C++ programming and also illustrate graphically the applicability of the new method. By getting the advantages of the mathematical building functions like pow (for power), exp (for exponential),…etc. are provided in C++ programming library, all processing steps are done efficiently and illustrated as Pseudocode model. This method enables us to approximate the solution as well as its first and third derivatives at every point of the range of integration. We proved that this new method gives better numerical results than the previous known results. In recent years, Al-Said and Noor [2, 3], Khalifa and Noor [4] and Noor and Al-Said [5, 6] have used such types of penalty function in solving a class of contact problems in elasticity in conjunction with collocation, finite difference and spline techniques. The general fourth order initial value problem considered is of the form functions, Pseudocode.

Canadian Journal of Physics, 1998

From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions. With mere Taylor expansions, Dirac delta functions and Fourier transform with its discrete counterpart, the DFT, we can obtain, from a digitized function, its integral between any limits, its Fourier transform without band limitations and its derivatives of any order. The same method intrinsically produces polynomial splines of any order and automatically generates the best possible end conditions. For a given digitized function, procedures to determine the optimum parameters of the method are presented. The way the method is structured makes it easy to estimate fairly accurately the error for any result obtained. Tests conducted on nontrivial numerical functions show that relative as well as absolute errors can be much smaller than 10-100, and there is no indication that even better results could n...

In first place, I am greatly indebted to my advisor, Emilio Defez, for his kind support and companionship, initiating my work in this area of mathematics and guiding the research for this thesis with him. Many thanks also go to my tutor, Antonio Hervás, for plenty of useful advise, and to Javier Ibáñez for his expertise on MATLAB in a fruitful collaboration with him.

In this paper, a kind of quasi-cubic Bézier curves by the blending of algebraic polynomials and trigonometric polynomials using weight method is presented, named WAT Bézier curves. Here weight coefficients are also shape parameters, which are called weight parameters. The interval [0, 1] of weight parameter values can be extended to [-2,π2/(π2-6)]. The WAT Bézier curves include cubic Bézier curves and C-Bézier curves () as special cases. Unlike the existing techniques based on C-Bézier methods which can approximate the Bézier curves only from single side, the WAT Bézier curves can approximate the Bézier curve from the both sides, and the change range of shape of the curves is wider than that of C-Bézier curves. The geometric effect of the alteration of this weight parameter is discussed. Some transcendental curves can be represented by the introduced curves exactly.

Computer-Aided Design, 2000

To describe the tool path of a CNC machine, it is often necessary to approximate curves by G 1 arc splines with the number of arc segments as small as possible. Ahn et al. have proposed an iterative algorithm for approximating quadratic Bézier curves by G 1 arc splines with fewer arc segments than the biarc method. This paper gives the formula of the upper bound for arc segments used by their algorithm. Based on the formula, two kinds of bisection algorithms for approximating quadratic Bézier curves by G 1 arc splines are presented. Results of some examples illustrate their efficiency. ᭧

International Journal of Mathematical Modelling & Computations, 2015

A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th. are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods and compared with the methods in [23,32-34].

International Journal of Electrical and Computer Engineering (IJECE), 2020

In this paper, a new method for the approximation of offset curves is presented using the idea of the parallel derivative curves. The best uniform approximation of degree 3 with order 6 is used to construct a method to find the approximation of the offset curves for Bezier curves. The proposed method is based on the best uniform approximation, and therefore; the proposed method for constructing the offset curves induces better outcomes than the existing methods.

PLOS ONE, 2019

An imperative curve modeling technique has been established with a view to its applications in various disciplines of science, engineering and design. It is a new spline method using piecewise quadratic trigonometric functions. It possesses error bounds of order 3. The proposed curve model also owns the most favorable geometric properties. The proposed spline method accomplishes C 2 smoothness and produces a Quadratic Trigonometric Spline (QTS) with the view to its applications in curve design and control. It produces a C 2 quadratic trigonometric alternative to the traditional cubic polynomial spline (CPS) because of having four control points in its piecewise description. The comparison analysis of QTS and CPS verifies the QTS as better alternate to CPS. Also, the time analysis proves QTS computationally efficient than CPS.

Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1972

Mathematics and Statistics, 2021

The Bezier curve is a parametric curve used in the graphics of a computer and related areas. This curve, connected to the polynomials of Bernstein, is named after the design curves of Renault's cars by Pierre Bézier in the 1960s. There has recently been considerable focus on finding reliable and more effective approximate methods for solving different mathematical problems with differential equations. Fuzzy differential equations (known as FDEs) make extensive use of various scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under uncertainty. This article discusses the use of Bezier curves for solving elevated order fuzzy initial value problems (FIVPs) in the form of ordinary differential equation. A Bezier curve approach is analyzed and updated with concepts and properties of the fuzzy set theory for solving fuzzy linear problems. The control points on Bezier curve are obtained by minimizing the residual function based on the least square method. Numerical examples involving the second and third order linear FIVPs are presented and compared with the exact solution to show the capability of the method in the form of tables and two dimensional shapes. Such findings show that the proposed method is exceptionally viable and is straightforward to apply.

Applied Mathematics and Computation, 2011

The main goal of the paper is to introduce methods which compute Bézier curves faster than Casteljau's method does. These methods are based on the spectral factorization of a n × n Bernstein matrix, B e n (s) = P n G n (s)P −1 n , where P n is the n × n lower triangular Pascal matrix. So we first calculate the exact optimum positive value t in order to transform P n in a scaled Toeplitz matrix, which is a problem that was partially solved by X. Wang and J. Zhou (2006). Then fast Pascal matrixvector multiplications and strategies of polynomial evaluation are put together to compute Bézier curves. Nevertheless, when n increases, more precise Pascal matrixvector multiplications allied to affine transformations of the vectors of coordinates of the control points of the curve are then necessary to stabilize all the computation.

International Journal of Autonomic Computing, 2009

Numerical methods for solving the system of linear algebraic equations as well as the system of differential equations have been known since the last century. Most numerical methods are very accurate and fast. However, some complicated problems can occur, such as stiff problems and bad-conditional equations, which could be computationally intensive. This paper describes the Modern Taylor Series Method for solving the system of algebraic equations using differential equations. The example of an electrical circuit with a parasitic capacity for demonstrating computational problems will be shown and the suggestion solution presented.

The Computer Journal, 1966

A spline function is a piecewise polynomial of degree m joined smoothly so that it has in -\ continuous derivatives. When used as an approximating function the spline provides a smooth yet flexible curve of relatively low degree.