Degree reduction of Bézier curves

Eck, Matthias

doi:10.1016/0167-8396(93)90039-6

Cited by 89 publications

(55 citation statements)

References 5 publications

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“…Our proof is also direct in the sense that in contrast to [Eck93] we do not assume that a relation of the form (6) is valid. Furthermore, our derivation implies a geometric interpretation of the situation that has not yet been given: the Bezier points of the optimal component-wise degree reduction of x lie on parallel lines with the direction vector 6. nb 0 through the Bezier points of a Taylor expansion of x of degree n -1.…”

Section: Introductionmentioning

confidence: 87%

“…Note, that in contrast to [Eck93] we do not assume that a relation of the form (6) holds. for all polynomials p E IIn with leading coefficient 1 it follows that…”

Section: Optimal Component-wise Degree Reduction In the Uniform Normmentioning

confidence: 99%

“…Formula (11) is equivalent to (6) because (12) but it reveals geometric information that has not been given ( and incorrectly depicted in fig.l, p.242) in [Eck93]: The Bezier points of the optimal component-wise degree reduction with respect to the uniform norm lie on parallel lines through the point b{ with direction 6nbo.…”

Section: Optimal Component-wise Degree Reduction In the Uniform Normmentioning

confidence: 99%

“…In early papers on the issue of degree reduction [For72], [Far83] non-optimal (but easy to compute) component-wise degree reductions to arbitrary Bezier curves have been constructed using simple convex combinations of the points b{ and b{ r. Optimal componentwise degree reductions have been considered in [Eck93] for the case of the uniform norm.…”

Section: Introductionmentioning

confidence: 99%

“…For an arbitrary Bezier curve x the points b{ and b[I defined by (2) and (3) or explicitly (cf [Eck93]) by…”

Section: Introductionmentioning

confidence: 99%

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The geometry of optimal degree reduction of Bézier curves

Brunnett

Schreiber

Braun

1996

Computer Aided Geometric Design

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.Optimal degree reductions, i.e. best approximations of n-th degree Bezier curves by Bezier curves of degree n -1, with respect to different norm:; are studied. lt is shown that for any Lp-norm the euclidean degree reduction where the norm is applied to the euclidean distance function of two curves is identical to componentwise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the B~zier points of any Taylor expansion of degree n -1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bezier points of the degree reduction are explicitly given in the unconstrained case for p = 1 and p = 2 and in the constrained case for p = 2.

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Section: Introductionmentioning

confidence: 87%

“…Note, that in contrast to [Eck93] we do not assume that a relation of the form (6) holds. for all polynomials p E IIn with leading coefficient 1 it follows that…”

Section: Optimal Component-wise Degree Reduction In the Uniform Normmentioning

confidence: 99%

Section: Optimal Component-wise Degree Reduction In the Uniform Normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For an arbitrary Bezier curve x the points b{ and b[I defined by (2) and (3) or explicitly (cf [Eck93]) by…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The geometry of optimal degree reduction of Bézier curves

Brunnett

Schreiber

Braun

1996

Computer Aided Geometric Design

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Least Squares Approximation of Bézier Coefficients Provides Best Degree Reduction in the L2-Norm

Peters

Reif

2000

Journal of Approximation Theory

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Approximate μ-Bases of Rational Curves and Surfaces

Shen

Chen

Jüttler

et al. 2006

Geometric Modeling and Processing - GMP 2006

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The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and surfaces. However, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and processing applications. In this paper, we introduce approximate μ-bases for rational curves and surfaces, and present an algorithm to compute approximate μ-bases. The algorithm amounts to solving a generalized eigenvalue problem and some quadratic programming problems with linear constraints. As applications, approximate implicitization and degree reduction of rational curves and surfaces with approximate μ-bases are discussed. Both the parametric equations and the implicit equations of the approximate curves/surfaces are easily obtained by using the approximate μ-bases. As indicated by the examples, the proposed algorithm may be a useful alternative to other methods for approximate implicitization.

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Degree reduction of Bézier curves

Cited by 89 publications

References 5 publications

The geometry of optimal degree reduction of Bézier curves

The geometry of optimal degree reduction of Bézier curves

Least Squares Approximation of Bézier Coefficients Provides Best Degree Reduction in the L2-Norm

Approximate μ-Bases of Rational Curves and Surfaces

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