Constrained polynomial degree reduction in the L2-norm equals best weighted Euclidean approximation of Bézier coefficients

Ahn, Young Joon; Lee, Byung-Gook; Park, Yunbeom; Yoo, Jaechil

doi:10.1016/j.cagd.2003.10.001

Cited by 49 publications

(46 citation statements)

References 17 publications

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“…They proved that the least squares approximation of Bézier coefficients provides the best polynomial degree reduction in the L 2 -norm. Two interesting generalizations of this result were achieved by Ahn et al in [2] and by Ait-Haddou in [3]. Ahn et al in [2] showed that a weighted least squares approximation of Bézier coefficients provides the best constrained polynomial degree reduction in the L 2 -norm.…”

Section: Introductionmentioning

confidence: 89%

“…Two interesting generalizations of this result were achieved by Ahn et al in [2] and by Ait-Haddou in [3]. Ahn et al in [2] showed that a weighted least squares approximation of Bézier coefficients provides the best constrained polynomial degree reduction in the L 2 -norm. By constrained we understand that the original polynomial and its reduced-degree approximation match at the boundaries up to a specific continuity order.…”

Section: Introductionmentioning

confidence: 89%

“…Optimal degree reduction is one of the fundamental tasks in Computer Aided Geometric Design (CAGD) and therefore has attracted researchers' attention for several decades [4,17,2,1,10,13,15]. Used not only for data compression, CAD/CAM software typically requires algorithms capable of converting a curve (surface) of a high degree to a curve (surface) of a lower degree.…”

Section: Introductionmentioning

confidence: 99%

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Constrained multi-degree reduction with respect to Jacobi norms

Ait‐Haddou

Bartoň

2016

Computer Aided Geometric Design

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Please cite this article in press as: Ait-Haddou, R., Bartoň, M. Constrained multi-degree reduction with respect to Jacobi norms. Comput. Aided Geom. Des. (2015), http://dx.doi.org/10. 1016/j.cagd.2015.12.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights• We study the problem of constrained degree reduction with respect to Jacobi norms.• Our results generalize several previous findings on polynomial degree reduction.• We explore the space of Jacobi parameters on the reduced polynomial approximation.Constrained multi-degree reduction with respect to Jacobi norms AbstractWe show that a weighted least squares approximation of Bézier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. This result affords generalizations to many previous findings in the field of polynomial degree reduction. A solution method to the constrained multi-degree reduction with respect to the Jacobi L 2 -norm is presented.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

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Constrained multi-degree reduction with respect to Jacobi norms

Ait‐Haddou

Bartoň

2016

Computer Aided Geometric Design

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“…However, when using the Bézier method to describe complex shapes, the problem of smooth joining still needs to be solved. The problem of continuity has also been studied in the literature when considering the degree reduction of Bézier curves with boundary constraints [14,15]. In general, the higher the requirement for smoothness is, the more complex the smooth joining condition is.…”

Section: Introductionmentioning

confidence: 99%

Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Yan

2016

MCA

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This paper aims to simplify the continuity conditions of Bézier curves. For this purpose, a special family of Bézier curves with three parameters, to be called adjustable Bézier curves, is constructed. They have the same structure as the quartic Bézier curves. The newly constructed curves possess some of the basic properties of Bézier curves, such as the convex hull property, symmetry, geometric invariance, etc., and they have shape adjustability. Moreover, under the geometric continuity of order 1 (G 1 ) conditions of the usual Bézier curves, the adjustable Bézier curves can reach geometric continuity of order k (G k ); here, k is one of the parameters of the newly constructed curves. The recursive evaluation algorithm of the new curves is provided. We also discuss how to construct the adjustable Bézier curves with a given tangent polygon. Numerical examples illustrate the correctness and validity of the proposed method.

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“…al. in [1], and discrete cases have been studied in [2], [8]. The existing methods to find degree reduction have many issues including accumulate round-off errors, stability issues, complexity, accuracy, losing conjugacy, requiring the search direction to be set to the steepest descent direction frequently, experiencing ill-conditioned systems, leading to a singularity, and the most challenging difficulty is in applying the methods (difficulty and indirect).…”

Section: Introductionmentioning

confidence: 99%

Weighted G1-Multi-Degree Reduction of B´ezier Curves

Rababah¹,

Ibrahim²

2016

ijacsa

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Abstract-In this paper, weighted G 1 -multi-degree reduction of Bézier curves is considered. The degree reduction of a given Bézier curve of degree n is used to write it as a Bézier curve of degree m, m < n. Exact degree reduction is not possible, and, therefore, approximation methods are used. The weight function w(t) = 2t(1 − t), t ∈ [0, 1] is used with the L2-norm in multidegree reduction with G 1 -continuity at the end points of the curve. Since we consider boundary conditions this weight function improves approximation in the middle of the curve. Numerical results and comparisons show that the proposed method produces fewer errors and outperform all existing methods.

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Constrained polynomial degree reduction in the L2-norm equals best weighted Euclidean approximation of Bézier coefficients

Cited by 49 publications

References 17 publications

Constrained multi-degree reduction with respect to Jacobi norms

Constrained multi-degree reduction with respect to Jacobi norms

Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Weighted G1-Multi-Degree Reduction of B´ezier Curves

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