Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials

Woźny, Paweł; Lewanowicz, Stanisław

doi:10.1016/j.cagd.2009.01.006

Cited by 56 publications

(42 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[6] [8] and [12] are mainly about G -continuity at the endpoints of the curve are derived in [5]. Two types of geometric constraints are presented in [7].…”

Section: Description About Degree Reduction Of Bézier Curvesmentioning

confidence: 99%

“…Figure 2 shows the constraint (13). The red one is the degree reduced curve derived from the conventional optimal function of Equation (6). The green and blue ones are obtained using the modified method of Equation (8).…”

Section: Graphic Experimentsmentioning

confidence: 99%

“…These researches can be classified by norm which the distance between polynomials is measured in, such as L ∞ -norm [2] [3], L 1 -norm [4], 2 L -norm [5] [8], or p L -norm [9] [10]. And the constrained degree reduction of Bézier curves with different parametric [6] [11] and geometric [5] [7] [12] continuity attend points have been studied in many previous papers.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

Han¹,

Yang²

2016

JAMP

View full text Add to dashboard Cite

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

show abstract

“…[6] [8] and [12] are mainly about G -continuity at the endpoints of the curve are derived in [5]. Two types of geometric constraints are presented in [7].…”

Section: Description About Degree Reduction Of Bézier Curvesmentioning

confidence: 99%

Section: Graphic Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

Han¹,

Yang²

2016

JAMP

View full text Add to dashboard Cite

show abstract

“…For extensive lists of references, see the recent papers of Lu [6], or Rababah and Mann [12]. The conventional problem of degree reduction differs from Problem 1.1 in considering, instead of condition (i), the C k,l -continuity at the endpoints of curves, i.e., R In the past 30 years, many papers dealing with this problem have been published (see, e.g., [2,3,13,14,15]). …”

Section: Introductionmentioning

confidence: 99%

“…In particular, in [15], two of us have proposed a method based on the use of the so-called dual Bernstein polynomials, which has complexity O(mn), the least among the existing algorithms. In the present paper, we apply an extended version of this method as an essential part of the algorithms of solving Problem 1.1.…”

Section: Introductionmentioning

confidence: 99%

G k,l -constrained multi-degree reduction of Bézier curves

2015

Self Cite

View full text Add to dashboard Cite

We present a new approach to the problem of G k,l -constrained (k, l ≤ 3) multi-degree reduction of Bézier curves with respect to the least squares norm. First, to minimize the least squares error, we consider two methods of determining the values of geometric continuity parameters. One of them is based on quadratic and nonlinear programming, while the other uses some simplifying assumptions and solves a system of linear equations. Next, for prescribed values of these parameters, we obtain control points of the multi-degree reduced curve, using the properties of constrained dual Bernstein basis polynomials. Assuming that the input and output curves are of degree n and m, respectively, we determine these points with the complexity O(mn), which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms.

show abstract

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

Jani

Babolian

Javadi

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the non-orthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein-spectral Petrov-Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.

show abstract

Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials

Cited by 56 publications

References 21 publications

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

G k,l -constrained multi-degree reduction of Bézier curves

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

Contact Info

Product

Resources

About

Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials

Cited by 56 publications

References 21 publications

Multi-Degree Reduction of B&#233;zier Curves with Distance and Energy Optimization

Multi-Degree Reduction of B&#233;zier Curves with Distance and Energy Optimization

G k,l -constrained multi-degree reduction of Bézier curves

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

Contact Info

Product

Resources

About

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization