Jacobi-Bernstein Basis Transformation

2004 ◽  
Vol 4 (2) ◽  
pp. 206-214 ◽  
Author(s):  
Abedallah Rababah
Keyword(s):  
Least Squares ◽  
Jacobi Polynomial ◽  
Bernstein Polynomial ◽  
Jacobi Polynomials ◽  
Polynomial Basis ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
The Matrix ◽  
Bernstein Polynomial Basis

Abstract In this paper we derive the matrix of transformation of the Jacobi polynomial basis form into the Bernstein polynomial basis of the same degree n and vice versa. This enables us to combine the superior least-squares performance of the Jacobi polynomials with the geometrical insight of the Bernstein form. Application to the inversion of the Bézier curves is given.

Transformation of Chebyshev–Bernstein Polynomial Basis

2003 ◽  
Vol 3 (4) ◽  
pp. 608-622 ◽  
Author(s):  
Abedallah Rababah
Keyword(s):  
Least Squares ◽  
Chebyshev Polynomials ◽  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
Polynomial Basis ◽  
Bernstein Basis ◽  
Linear Maps ◽  
Bernstein Polynomial Basis ◽  
The Stability

AbstractIn this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite, power-Hermite, and Bernstein–Legendre basis transformations.

scholarly journals Jacobi-weighted orthogonal polynomials on triangular domains

2005 ◽  
Vol 2005 (3) ◽  
pp. 205-217 ◽  
Author(s):  
A. Rababah ◽  
M. Alqudah
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Bernstein Polynomial ◽  
Orthogonal System ◽  
Polynomial Basis ◽  
Bernstein Basis ◽  
Triangular Domain ◽  
Jacobi Weight ◽  
Bernstein Polynomial Basis ◽  
Alpha Beta

We construct Jacobi-weighted orthogonal polynomials𝒫n,r(α,β,γ)(u,v,w),α,β,γ>−1,α+β+γ=0, on the triangular domainT. We show that these polynomials𝒫n,r(α,β,γ)(u,v,w)over the triangular domainTsatisfy the following properties:𝒫n,r(α,β,γ)(u,v,w)∈ℒn,n≥1,r=0,1,…,n,and𝒫n,r(α,β,γ)(u,v,w)⊥𝒫n,s(α,β,γ)(u,v,w)forr≠s. And hence,𝒫n,r(α,β,γ)(u,v,w),n=0,1,2,…,r=0,1,…,nform an orthogonal system over the triangular domainTwith respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
M. A. Saker
Keyword(s):  
Jacobi Polynomials ◽  
Basis Functions ◽  
Least Square ◽  
Natural Basis ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided Geometric Design ◽  
Curves And Surfaces ◽  
Endpoint Constraints

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

Solution of the Pellet Equation by use of the Orthogonal Collocation and Least Squares Methods: Effects of Different Orthogonal Jacobi Polynomials

2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Jannike Solsvik ◽  
Hugo Jakobsen
Keyword(s):  
Numerical Methods ◽  
Least Squares ◽  
Collocation Method ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Least Squares Method ◽  
Condition Numbers ◽  
Orthogonal Collocation ◽  
Point Distribution ◽  
Boundary Points

Two numerical methods in the family of weighted residual methods; the orthogonal collocation and least squares methods, are used within the spectral framework to solve a linear reaction-diffusion pellet problem with slab and spherical geometries. The node points are in this work taken as the roots of orthogonal polynomials in the Jacobi family. Two Jacobi polynomial parameters, alpha and beta, can be used to tune the distribution of the roots within the domain. Further, the internal points and the boundary points of the boundary-value problem can be given according to: i) Gauss-Lobatto-Jacobi points, or ii) Gauss-Jacobi points plus the boundary points. The objective of this paper is thus to investigate the influence of the distribution of the node points within the domain adopting the orthogonal collocation and least squares methods. Moreover, the results of the two numerical methods are compared to examine whether the methods show the same sensitivity and accuracy to the node point distribution. The notifying findings are as follows: i) The Legendre polynomial, i.e., alpha=beta=0, is a very robust Jacobi polynomial giving the better condition number of the coefficient matrix and the polynomial also give good behavior of the error as a function of polynomial order. This polynomial gives good results for small and large gradients within both slab and spherical pellet geometries. This trend is observed for both of the weighted residual methods applied. ii) Applying the least squares method the error decreases faster with increasing polynomial order than observed with the orthogonal collocation method. However, the orthogonal collocation method is not so sensitive to the choice of Jacobi polynomial and the method also obtains lower error values than the least squares method due to favorable lower condition numbers of the coefficient matrices. Thus, for this particular problem, the orthogonal collocation method is recommended above the least squares method. iii) The orthogonal collocation method show minor differences between Gauss-Lobatto-Jacobi points and Gauss-Jacobi plus boundary points.

scholarly journals Using Bezier curves in medical applications

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 937-943 ◽  
Author(s):  
Buket Simsek ◽  
Ahmet Yardimci
Keyword(s):  
Cross Sections ◽  
Biomedical Applications ◽  
Medical Physics ◽  
Binomial Coefficients ◽  
Medical Applications ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.

scholarly journals Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 967 ◽  
Author(s):  
Samia BiBi ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro
Keyword(s):  
Geometric Modeling ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curvature Continuity ◽  
Bernstein Basis Functions

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

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