A note on Class A Bézier curves

Cao, Juan; Wang, Guozhao

doi:10.1016/j.cagd.2007.10.001

Cited by 30 publications

(19 citation statements)

References 1 publication

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“…3D class A Bézier curves are originally proposed by Farin [4]. For a symmetric matrix M , Cao et al have presented a method for checking the monotonicity of the curvature and torsion [1]. Matrix M , which we use in this paper, is not symmetric in general.…”

Section: D Class a Bézier Curvesmentioning

confidence: 99%

Interactive Generation of 3D Class A Bézier Curve Segments

Yoshida

Fukuda

Saito

2010

Computer-Aided Design and Applications

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This paper presents a method for interactively generating a 3D class A Bézier curve segment by specifying two endpoints and their tangents. We clarify geometric properties of 3D class A Bézier curves and use them for efficiently generating 3D class A Bézier curve segments satisfying the specified positional and tangential constraints. The characteristics of typical 3D class A Bézier curves are also clarified. 164 segments [14] can be interactive controlled with user-specified endpoint constraints. However, the curve is not compatible with free-form curves, such as Bézier curves or NURBS curves.

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Section: D Class a Bézier Curvesmentioning

confidence: 99%

Interactive Generation of 3D Class A Bézier Curve Segments

Yoshida

Fukuda

Saito

2010

Computer-Aided Design and Applications

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“…In 2014, Chu [9] used Beta functions to construct a kind of Beta-Bézier basis function, and thereby give the corresponding definition of the Beta-Bézier curve with one shape parameter. Furthermore, Farin et al [10] and Cao et al [11] gave two different kinds of high-degree Bézier curves with single shape parameters, respectively. As only one shape parameter is contained in curves in the literature [6][7][8][9][10][11], its value can be modified to make this type of curve merely simply swing from one or two sides of classical Bézier curves.…”

Section: Of 23mentioning

confidence: 99%

“…To sum up, the local shape adjustments of the curves and surfaces in [6][7][8][9][10][11] are difficult and caused by the absence of a local shape parameter, while the global shapes adjustment in [12][13][14][15][16][17][18][19][20][21][22] are inconvenient as a result of the lack of a global shape parameter. In practical engineering applications, the local or global shapes of curves or surfaces often need to be modified under the condition of the control points remaining the same.…”

Section: Of 23mentioning

confidence: 99%

“…Hu et al introduced certain polynomial curves and surfaces called SG-Bézier curves, which can be regarded as an extension of classic Bézier curves; their global and local shape can be adjusted by altering the global and local shape parameters under the condition of keeping the control points unchanged, respectively [23]. The theoretical analysis shows that the proposed SG-Bézier curves hold most of the advantages of the classical Bézier model, as well as overcomes the disadvantages of the Bézier model [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Of 23mentioning

confidence: 99%

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Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities

et al. 2018

Symmetry

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The modeling of free-form engineering complex curves is an important subject in product modeling, graphics, and computer aided design/computer aided manufacturing (CAD/CAM). In this paper, we propose a novel method to construct free-form complex curves using shape-adjustable generalized Bézier (or SG-Bézier, for short) curves with constraints of geometric continuities. In order to overcome the difficulty that most of the composite curves in engineering cannot often be constructed by using only a single curve, we propose the necessary and sufficient conditions for G1 and G2 continuity between two adjacent SG-Bézier curves. Furthermore, the detailed steps of smooth continuity for two SG-Bézier curves, and the influence rules of shape parameters on the composite curves, are studied. We also give some important applications of SG-Bézier curves. The modeling examples show that our methods in this paper are very effective, can easily be performed, and can provide an alternative powerful strategy for the design of complex curves.

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“…More recently, Farin has proposed class A Bézier curves [4]. Cao and Wang presented the correct condition for class A Bézier curves generated by symmetric matrices [1]. The method for interactively drawing class A Bézier curve segments is not known except for the curve is a typical curve.…”

Section: Related Workmentioning

confidence: 99%

Interactive Control of Planar Class A Bézier Curves using Logarithmic Curvature Graphs

Yoshida¹,

Hiraiwa²,

Saito³

2008

Computer-Aided Design and Applications

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We present a method for interactively drawing a planar class A Bézier curve segment. First, we present a method for interactively drawing a typical class A Bézier curve segment by specifying three points like a quadratic Bézier curve segment. We show that as the degree of a typical class A Bézier curve segment is elevated, the curve converges to a logarithmic spiral segment. At the limit of infinite degree, the curve segment becomes a logarithmic spiral segment. We also present a method for drawing a general class A Bézier curve segment by perturbing the elements of the typical class A matrix so that the endpoint constraints are satisfied. To see the characteristics of the generated curves, we propose to use logarithmic curvature graphs.

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A note on Class A Bézier curves

Cited by 30 publications

References 1 publication

Interactive Generation of 3D Class A Bézier Curve Segments

Interactive Generation of 3D Class A Bézier Curve Segments

Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities

Interactive Control of Planar Class A Bézier Curves using Logarithmic Curvature Graphs

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