scholarly journals Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghomanjani ◽  
A. Kılıçman ◽  
S. Effati
Keyword(s):  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Control Points ◽  
Volterra Type ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using Bezier curves. The Bezier curves are chosen as piecewise polynomials of degreen, and Bezier curves are determined on[t0, tf ]by n+1control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.

scholarly journals Bezier Curves for Solving Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
F. Ghomanjani ◽  
M. H. Farahi ◽  
A. Kılıçman
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Direct Algorithm ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

The Bezier curves are presented to estimate the solution of the linear Fredholm integral equation of the second kind. A direct algorithm for solving this problem is given. We have chosen the Bezier curves as piecewise polynomials of degreenand determine Bezier curves on [0, 1] byn+1control points. Numerical examples illustrate that the algorithm is applicable and very easy to use.

scholarly journals Geometric Piecewise Cubic Bézier Interpolating Polynomial with C2 Continuity

2021 ◽  
Vol 20 (1) ◽  
pp. 133-159
Author(s):  
Mustafa Fadhel ◽  
Zurni Omar
Keyword(s):  
Internal Control ◽  
Missing Values ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Piecewise Polynomials ◽  
Partial Smoothness ◽  
C2 Continuity ◽  
Interpolation Polynomials ◽  
The Given

Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods.

scholarly journals Quasi-Bézier Curves with Shape Parameters

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jun Chen
Keyword(s):  
Convex Hull ◽  
Basis Functions ◽  
Geometrical Shape ◽  
Control Points ◽  
Shape Parameters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Universal Form ◽  
Special Cases

The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding Quasi-Bézier curves were constructed step-by-step in this paper, using the method of undetermined coefficients. The series of Quasi-Bézier curves had geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints and tangent edges at the endpoints, and shape adjustability while maintaining the control points. Various existing Quasi-Bézier curves became special cases in the series. The obvious geometric significance of shape parameters made the adjustment of the geometrical shape easier for the designer. The numerical examples indicated that the algorithm was valid and can easily be applied.

scholarly journals Bezier Curves Method for Fourth-Order Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
F. Ghomanjani ◽  
A. V. Kamyad ◽  
A. Kılıçman
Keyword(s):  
Fourth Order ◽  
Necessary Conditions ◽  
Integrodifferential Equations ◽  
Lagrange Equations ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Calculus Of Variation ◽  
Numerical Examples ◽  
Linear And Nonlinear

The Bezier curves method is applied to solve both linear and nonlinear BVPs for fourth-order integrodifferential equations. Also, the presented method is developed for solving BVPs which arise from the problems in calculus of variation. These BVPs result from the Euler-Lagrange equations which are the necessary conditions of the extremums of problems in calculus of variation. Some numerical examples demonstrate the validity and applicability of the technique.

scholarly journals Bezier Curves Based Numerical Solutions of Delay Systems with Inverse Time

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
F. Ghomanjani ◽  
M. H. Farahi ◽  
A. Kılıçman ◽  
A. V. Kamyad ◽  
N. Pariz
Keyword(s):  
Numerical Solutions ◽  
Delay Systems ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Delay Function ◽  
Piecewise Polynomials ◽  
Delay Differential ◽  
First Time ◽  
Present Algorithm

This paper applied, for the first time, the Bernstein’s approximation on delay differential equations and delay systems with inverse delay that models these problems. The direct algorithm is given for solving this problem. The delay function and inverse time function are expanded by the Bézier curves. The Bézier curves are chosen as piecewise polynomials of degreen, and the Bézier curves are determined on any subinterval byn+1control points. The approximated solution of delay systems containing inverse time is derived. To validate accuracy of the present algorithm, some examples are solved.

Smoothness of C-Bezier Curves

2000 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Shape Control ◽  
Control Point ◽  
Design Procedure ◽  
Design Parameters ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Point Distribution ◽  
G1 Continuity ◽  
G2 Continuity

Abstract This paper presents G1 and G2 continuity conditions of c-Bezier curves. It shows that the collinear condition for G1 continuity of Bezier curves is generally no longer necessary for c-Bezier curves. Such a relaxation of constraints on control points is beneficial from the structure of c-Bezier curves. By using vector weights, each control point has two extra free design parameters, which offer the probability of obtaining G1 and G2 continuity by only adjusting the weights if the control points are properly distributed. The enlargement of control point distribution region greatly simplifies the design procedure to and enhances the shape control on constructing composite curves.

scholarly journals Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

Style Design System Based on Class A Bézier Curves for High-Quality Aesthetic Shapes

2012 ◽  
Author(s):  
Tetsuo Oya ◽  
Fumihiko Kimura ◽  
Hideki Aoyama
Keyword(s):  
Design System ◽  
Design Tools ◽  
Control Points ◽  
Modeling Framework ◽  
High Quality ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Class A ◽  
Monotone Curvature ◽  
Curvature And Torsion

In this paper, a style design system in which the conditions for Class A Bézier curves are applied is presented to embody designer’s intention by aesthetically high-quality shapes. Here, the term “Class A” means a high-quality shape that has monotone curvature and torsion, and the recent industrial design requires not only aesthetically pleasing aspect but also such high-quality shapes. Conventional design tools such as normal Bézier curves can represent any shapes in a modeling system; however, the system only provides a modeling framework, it does not necessarily guarantee high-quality shapes. Actually, designers do a cumbersome manipulation of many control points during the styling process to represent outline curves and feature curves; this hardship prevents designers from doing efficient and creative styling activities. Therefore, we developed a style design system to support a designer’s task by utilizing the Class A conditions of Bézier curves with monotone curvature and torsion.

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