Besides inheriting the properties of classical Bézier curves of degree
Bézier curves are widely used in Computer Aided Geometric Design (CAGD) and computer graphics (CG) which have many properties that are helpful for shape design. Developing more convenient techniques for designing and modifying Bézier curves is an important problem in computer aided design (CAD), computer aided manufacturing (CAM), and NC technology fields; see [
Many efforts have been made to develop more convenient and effective methods for shape modification of parametric curves and surfaces. Piegl [
Bézier curves have now become a powerful tool for constructing free-form curves in CAD/CAM. However, the Bézier model is imperfect, and it has its own shortage. That is, after choosing the basis functions, the shape of a Bézier curve is well determined by its control points. In the recent years, in order to overcome the shortage of Bézier curves, many scholars constructed new curves whose structures are similar to the Bézier curves by introducing parameters into basis functions; see [
The remainder of the paper is organized as follows. The definition and properties of
The definition of extension Bernstein basis functions is given as follows [
Let
For any integer
The extension Bernstein basis functions of degree n can be expressed explicitly as
The extension Bernstein basis functions (
Given control points
From the properties of the extension Bernstein basis functions (
And the derivative at end-points will satisfy
Furthermore, the second derivative at end-points can be represented as
A Bézier curve is defined as a parametric one which forms the basis of the Bernstein function. However, once the control points and their corresponding Bernstein polynomials are given, the shape of a Bézier curve is formed uniquely and there is no possibility to adjust it anymore. Modifying the shape of Bézier curves essentially requires the adjustments of vertexes of the control polygon, which is very inconvenient. For these reasons, the problem of shape modification of curves is proposed. Although the weights in NURBS method can adjust the shapes of NURBS curve and the NURBS curve has good properties and can express the conic section, the NURBS curve also has disadvantages, such as difficulty in choosing the value of the weight, the increased order of rational fraction caused by the derivation, and the need for a numerical method of integration.
The shape parameters are applied to generate some curves whose shape is adjustable as an extension of the existing method. The
To sum up, with the extra degree of freedom provided by the shape parameter
Performance comparisons of
Property | Bézier curves | NURBS curves | |
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Property of basis functions | |||
Nonnegativity |
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Partition of unity |
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Symmetry |
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Shape parameters |
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Linear independence |
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Degeneracy |
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Property of the Curves | |||
Variation diminishing property |
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Affine invariability |
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Convex hull property |
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Symmetry |
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End-point properties |
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Extra degree of freedom |
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Computational complexity | Low | Low | High |
Figure
Suppose the position vector of
For
When
Let perturbation
The control points can be modified by (
Let tangent vector of
For
According to (
Notice that
When
Letting
When
The control points can be modified according to (
Let position vector of
When
Setting
Denote
For
The control points can be modified by (
Let position vector of
When
Letting
Let
For
The control points can be modified by (
In this section, we will give three examples to show the effects of the proposed method.
Given shape parameter
Shape modification for quartic
Modification of control points
Modification of control points
Modification of control points
Shape modification for quartic
Modification of control points
Modification of control points
Shape modification for quartic
Modification of control points
Modification of control points
Figure
Shape modification for quartic
Modification of control points
Modification of control points
Given shape parameter
Shape modification for
Position vector constraint
Tangent vector constraint
Position vector and tangent vector constraints
Given shape parameter
Shape modification for
Position vector constraint
Tangent vector constraint
Position vector and tangent vector constraints
In this paper, we give the definition of
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the referees for their helpful suggestions and comments which have improved the paper. This work is supported by the National Natural Science Foundation of China (nos. 51305344 and 11426173). This work is also supported by the Research Fund of Department of Science and Department of Education of Shaanxi, China (no. 2013JK1029).