Application of Legendre–Bernstein basis transformations to degree elevation and degree reduction

Lee, Byung-Gook; Park, Yunbeom; Yoo, Jaechil

doi:10.1016/s0167-8396(02)00164-4

Cited by 40 publications

(14 citation statements)

References 9 publications

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“…In general, degree reduction cannot be done exactly so that it invokes approximation problems. Thus in recent twenty years, many works [1][2][3][4][6][7][8][13][14][15][16]18,20,21,23] relevant to the degree reduction of Bézier curves have been published.…”

Section: Introductionmentioning

confidence: 99%

“…The Chebyshev polynomials of the first kind are used to obtain the best approximation in L ∞ -norm [7,14,23] and in the 1/ √ 4t − 4t 2 -weighted square norm [18], whereas the Chebyshev polynomials of the second kind are used in L 1 -norm [13]. The Legendre polynomials are used in L 2 -norm [8,14,15]. And the Jacobi polynomials are used in L ∞ -norm [1], in L 1 -norm [6], in L 2 -norm [4,20], and in L p -norm [3].…”

Section: Introductionmentioning

confidence: 99%

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Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Wang

2008

Journal of Computational and Applied Mathematics

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A polynomial curve on [0, 1] can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind. We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other, and examine the stability of this linear map. In the p = 1 and ∞ norms, the condition number of the Chebyshev-Bernstein transformation matrix grows at a significantly slower rate with n than in the power-Bernstein case, and the rate is very close (somewhat faster) to the Legendre-Bernstein case. Using the transformation matrices, we present a method for the best multi-degree reduction with respect to the t − t 2 -weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multi-degree reduction with constraints of endpoints continuity of orders r, s (r, s ≥ 0). This method has a quadratic complexity, and may be ill-conditioned when it is applied to the curves of high degree. We estimate the posterior L 1 -error bounds for degree reduction.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Wang

2008

Journal of Computational and Applied Mathematics

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“…Thus many papers dealing with the degree reduction have been published in the recent 30 years. They are classified by different norm in which the distance between polynomials is measured, e.g., in L ∞ -norm [8,14], in L 2 -norm [16,17,20], in L 1 -norm [13] or in L p -norm [6,12], etc. Furthermore, the constrained degree reduction of Bézier curves with C a−1 -constraint at boundary points is developed in many previous literature [1,2,5,7,9,11,15,[18][19][20][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

Kim¹,

Ahn²

2008

Journal of Computational and Applied Mathematics

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In this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d-dimensional simplex domain with respect to the L 2 -inner product and the weighted Euclidean inner product of BB (Bézier-Bernstein) coefficients are equal. Using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.

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“…For example, the Legendre series is used to develop a curve offsetting algorithm [1] , and the transformation between the Legendre basis and the Bernstein basis is implement to the approximation of the inversion for polynomials in Bernstein form [2] . The Tchebyshev basis and the Legendre basis are also applied to degree reduction of the Bézier curves or the interval Bézier curves [3][4][5][6] . Why can the Tchebyshev basis and the Legendre basis be used widely?…”

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confidence: 99%

An orthogonal basis for the hyperbolic hybrid polynomial space

Huang

Wang

2007

SCI CHINA SER F

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Motivated by the wide usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bézier basis in the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bézier basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the HBézier curves is sketched to illustrate the utility of the orthogonal basis. H-Bézier basis, orthogonal basis, basis transformations, degree reductionThe orthogonal polynomials play a fundamental role in problems of the least square approximation of functions. The polynomial curves and surfaces in the Tchebyshev basis and the Legendre basis have found their applications in many geometric operations in CAGD and CAD/CAM, such as intersecting, offsetting and degree reduction. For example, the Legendre series is used to develop a curve offsetting algorithm [1] , and the transformation between the Legendre basis and the Bernstein basis is implement to the approximation of the inversion for polynomials in Bernstein form [2] . The Tchebyshev basis and the Legendre basis are also applied to degree reduction of the Bézier curves or the interval Bézier curves [3][4][5][6] . Why can the Tchebyshev basis and the Legendre basis be used widely? We find that all these operations are done in the Bézier and NURBS models. The two bases are the special parameter polynomials in Jacobi polynomials that are orthogonal and have some excellent properties. Furthermore, they both can be converted to the Bernstein form comparatively stable.The Bézier and NURBS curves have become standard models; however, they cannot encompass transcendent curves that are expressed in nonalgebraic functions, such as the circle, the cycloid, the catenary and the hyperbolic helix. Therefore, there are many researches developed concerning the blending polynomial spaces. For example, the C-Bézier curve is constructed in the algebraic and triangular functions space [7][8][9][10][11] , and the H-Bézier curve appears in the algebraic

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Application of Legendre–Bernstein basis transformations to degree elevation and degree reduction

Abstract: We study the relationship of transformations between Legendre and Bernstein basis. Using the relationship, we present a simple and efficient method for optimal multiple degree reductions of Bézier curves with respect to the L 2 -norm.

Cited by 40 publications

References 9 publications

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

An orthogonal basis for the hyperbolic hybrid polynomial space

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