A unified matrix representation for degree reduction of Bézier curves

Sunwoo, Hasik; Lee, Namyong

doi:10.1016/j.cagd.2003.07.007

Cited by 16 publications

(5 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Computationally efficient handling of complex shapes often requires optimal reduction of Bézier curves based on different metrics [45]- [47]. This motivates many alternative approaches for degree reduction of Bézier curves [48] and their approximate conversions [49] (with end point constraints [50]). This present paper brings such CAD tools to the motion planning literature with important additions which, we believe, also contribute back to the CAD literature.…”

Section: A Motivation and Related Literaturementioning

confidence: 99%

See 1 more Smart Citation

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

Arslan¹,

Tiemessen²

2022

Preprint

View full text Add to dashboard Cite

As a parametric polynomial curve family, Bézier curves are widely used in safe and smooth motion design of intelligent robotic systems from flying drones to autonomous vehicles to robotic manipulators. In such motion planning settings, the critical features of high-order Bézier curves such as curve length, distance-to-collision, maximum curvature/velocity/acceleration are either numerically computed at a high computational cost or inexactly approximated by discrete samples. To address these issues, in this paper we present a novel computationally efficient approach for adaptive approximation of high-order Bézier curves by multiple low-order Bézier segments at any desired level of accuracy that is specified in terms of a Bézier metric. Accordingly, we introduce a new Bézier degree reduction method, called parameterwise matching reduction, that approximates Bézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new Bézier metric, called the maximum control-point distance, that can be computed analytically, has a strong equivalence relation with other existing Bézier metrics, and defines a geometric relative bound between Bézier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed Bézier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an n th -order Bézier curve can be accurately approximated by 3(n − 1) quadratic and 6(n − 1) linear Bézier segments, which is fundamental for Bézier discretization.

show abstract

Section: A Motivation and Related Literaturementioning

confidence: 99%

“…Many existing notions of Bézier degree reduction methods that are defined in terms of different Bézier distances (possibly with end-point constraints) can be unified using the inverse of degree elevation[48].…”

mentioning

confidence: 99%

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

Arslan¹,

Tiemessen²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The second category can be summarized as the geometric approximation algorithms with the control vertexes. In this category, reference (Forrest A R 1972) is based on the interpolation algorithm; S M. Hu proposes an approximation algorithm with the degree reduction; Reference (Sunwoo H and Lee N 2004) is based on the generalized matrix reduction technique. In reference (Jiuping.…”

Section: Introductionmentioning

confidence: 99%

An Approximation Method of Bézier Curve

Wu¹,

Song²,

Bao³

2017

CIS

View full text Add to dashboard Cite

It is proved that the linear space constructed by power base is a banach space under 2-norm by using approximation method. For the Bézier curve--the elements in banach space, the linear combination of the low-order S power base is used to approximate optimal the high-order Bernstein base function. The original Bézier curve is instituted by the linear combination of low-order S power base and the optimal approximation element of the original Bézier curve is obtained.

show abstract

“…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%

$G^{k,l}$-constrained multi-degree reduction of Bézier curves

Gospodarczyk,

Lewanowicz,

Woźny

2015

Preprint

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

A unified matrix representation for degree reduction of Bézier curves

Cited by 16 publications

References 10 publications

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

An Approximation Method of Bézier Curve

$G^{k,l}$-constrained multi-degree reduction of Bézier curves

Contact Info

Product

Resources

About