Multiple degree elevation and constrained multiple degree reduction for DP curves and surfaces

Aphirukmatakun, Chanon; Dejdumrong, Natasha

doi:10.1016/j.camwa.2010.09.052

Cited by 8 publications

(5 citation statements)

References 5 publications

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“…The study of the Ball curve inspired many scholars to explore its accurate shape by increasing or decreasing its degree through theoretical computation [3,12,[15][16][17][18][19][20]. Hence, the generalized Ball curves extended to Wang Ball curves, Said Ball curves, DP Ball curves and rational Ball curves with higher degree 𝑛 polynomials [10,[12][13]16].…”

Section: Introductionmentioning

confidence: 99%

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

Bhatti¹,

Karim²

2022

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Numerical methods are regularly established for the better approximate solutions of the ordinary differential equations (ODEs). The best approximate solution of ODEs can be obtained by error reduction between the approximate solution and exact solution. To improve the error accuracy, the representations of Wang Ball curves are proposed through the investigation of their control points by using the Least Square Method (LSM). The control points of Wang Ball curves are calculated by minimizing the residual function using LSM. The residual function is minimized by reducing the residual error where it is measured by the sum of the square of the residual function of the Wang Ball curve's control points. The approximate solution of ODEs is obtained by exploring and determining the control points of Wang Ball curves. Two numerical examples of initial value problem (IVP) and boundary value problem (BVP) are illustrated to demonstrate the proposed method in terms of error. The results of the numerical examples by using the proposed method show that the error accuracy is improved compared to the existing study of Bézier curves. Successfully, the convergence analysis is conducted with a two-point boundary value problem for the proposed method.

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Section: Introductionmentioning

confidence: 99%

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

Bhatti¹,

Karim²

2022

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“…As the commonly-used parameterized representation, Bézier curve segment and surface patch have explicit geometric properties and well-behaved control within a convex hull for any degree [10][11][12][13]. Nowadays, many researches have been published about Bézier curve or surface representation, including arithmetic geometry, model representation, optimization algorithm, applied mathematics and approximation theories [14][15][16][17][18]. Moreover, more efforts have made on improvement on novel curves and their continuities, curvature properties, curve fitting, etc [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

An equivalent parameter geometric shape representation using independent coordinates of cubic Bézier control points

et al. 2022

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Bézier surface has been commonly applied to represent the complex geometric shape. Generally, all control points are dealt with by the same blending functions, regardless of the effect of independent coordinate. It causes to lack the modeling flexibility. Therefore, this paper proposes an equivalent parameter geometric shape representation method using the independent coordinates of control points. Since the coordinate components of control points are independent, the geometric modeling becomes more flexible. Firstly, a general Bézier curve is described in detail. Related expression is brought out in the form of independent coordinates by introducing two parameters. Then, their geometric meanings are analyzed in detail. Since both parameters are independent to parametric variables u and v, Bézier curve possess the same interval in the discrete parametric space, namely equivalent parameter. Next, a bicubic Bézier subsurface patch representation is discussed, including regular and non-regular subsurface patch. A general surface expression is given out in the form of independent coordinates, as well as the parameter structure and the geometric transformations. Finally, an example of ‘Bézier tree branch’ is constructed by using the proposed method. Results shows that the proposed method is feasible and reasonable.

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“…Itsariyawanich and Dejdumrong dealed with the degree reduction of DP curve by transforming DP curve into Bézier curve [11] . Aphirukmatakun and Dejdumrong derived single and multiple degree elevation matrices by using monomial matrix representations of the DP polynomials [12] . Liu and Wang give an effective algorithm of constrained multidegree reduction approximation based on Jacobi basis [13] .…”

Section: Introductionmentioning

confidence: 99%

Approximate Merging of a Pair of DP Curves

Cai

Cheng

2012

2012 Fourth International Conference on Computational and Information Sciences

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Multiple degree elevation and constrained multiple degree reduction for DP curves and surfaces

Cited by 8 publications

References 5 publications

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

An equivalent parameter geometric shape representation using independent coordinates of cubic Bézier control points

Approximate Merging of a Pair of DP Curves

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