The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding Quasi-Bézier curves were constructed step-by-step in this paper, using the method of undetermined coefficients. The series of Quasi-Bézier curves had geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints and tangent edges at the endpoints, and shape adjustability while maintaining the control points. Various existing Quasi-Bézier curves became special cases in the series. The obvious geometric significance of shape parameters made the adjustment of the geometrical shape easier for the designer. The numerical examples indicated that the algorithm was valid and can easily be applied.
The Bézier curve
Here, Bernstein basis functions
Given that the shape of the curve is characterized by the control polygon, the designer always adjusts the control point
The rational Bézier curve
By assigning a weight
In addition, the algebraic trigonometric/hyperbolic curve
The simple form of the algebraic trigonometric/hyperbolic curve
In view of the fact that the expression of the parametric curve is determined by the control points and the basis functions, the properties of such functions identify the properties of the curve with its fixed control points. Therefore, several kinds of polynomial basis functions with shape parameters [
By letting
For the sake of concision, the notations
With the extra degree of freedom provided by the shape parameters
Properties of the basis functions and the curves with shape parameters.
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Basis functions with multiple shape parameters | |||||||
Nonnegativity |
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Partition of unity |
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Symmetry |
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Multiple shape parameters |
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Linear independence |
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Degeneracy |
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Curve with multiple shape parameters | |||||||
Geometric and affine invariability |
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Convex hull property |
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Symmetry |
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Interpolation at the endpoints |
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Tangent at the end edge |
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*The property of symmetry in [
The construction of the basis functions with shape parameters is the key step in [ In all existing research, the basis functions with shape parameters are initially given, and whether or not these functions and the corresponding curves have inherited the characteristics of the Bernstein basis functions and the Bézier curve, respectively, is examined. However, the method of obtaining the complex expressions of the basis functions remains unclear. Are these basis functions obtained through intuition or through an aimless attempt? There are numerous known basis functions with shape parameters in varying forms. Is there a type of Quasi-Bernstein basis function, which makes existing basis functions with shape parameters be its special case?
To answer the previous two questions, this paper uses the method of undetermined coefficients, which clarifies the construction process of the Quasi-Bernstein basis functions. A series of Quasi-Bernstein basis functions are finally obtained, rendering the existing basis function with shape parameters as their special case.
First, the following vectors are introduced:
Equation (
Given that
Thus, as long as the elements in the matrix
The
The Quasi-Bernstein basis functions
A sufficient condition for
Here,
Clearly, there is no row with all elements being 0 in
The necessary and sufficient condition for
It is known that
According to the linear independence of the Bernstein basis functions
By combining (
The necessary and sufficient condition for
According to the symmetry of the Bernstein basis functions
According to the linear independence of the Bernstein basis functions
The necessary and sufficient condition for the linear independence of
It is known that
According to the linear independence of the Bernstein basis functions
When (
If the elements
When the elements
Comparing (
If
The necessary and sufficient condition for
Clearly, the necessary and sufficient condition for
It is known that
Thus, the necessary and sufficient condition for
Similarly, the necessary and sufficient condition for
It is known that
Thus, the necessary and sufficient condition for
Hence, the necessary and sufficient condition for
When
The necessary and sufficient condition for
It is known that
Clearly, the necessary and sufficient condition for
Similarly, the necessary and sufficient condition for
When
All shape parameter matrixes that satisfy (
Here,
In summary, the Quasi-Bézier curve shape adjustability: the shape of the Quasi-Bézier curve can still be adjusted by maintaining the control points. geometric invariability: the Quasi-Bézier curve only relies on the control points, whereas it has nothing to do with the position and direction of the coordinate system; in other words, the curve shape remains invariable after translation and revolving in the coordinate system; affine invariability: barycentric combinations are invariant under affine maps; therefore, ( symmetry: whether the control points are labeled which follows the inspection of ( convex hull property: this property exists since the Quasi-Bernstein basis functions interpolation at the endpoints and tangent edges at the endpoint: the Quasi-Bézier curve
According to (
The shape parameter matrix
The geometric significance of the shape parameters
Quasi-Bernstein basis functions and Quasi-Bézier curves when
The shape parameter matrix
The geometric significance of the shape parameters
If we increase the value of
Quasi-Bernstein basis functions and Quasi-Bézier curves when
The shape parameter matrix
The geometric significance of the shape parameters
Quasi-Bernstein basis functions and Quasi-Bézier curves when
The shape parameter matrix
The geometric significance of the shape parameters
If we increase the value of
Quasi-Bernstein basis functions and Quasi-Bézier curves when
Several Quasi-Bernstein basis functions for low degree and low order are presented aforementioned. The corresponding basis functions for higher degree and higher order are defined recursively as follows [
Here, we set
Figure
Three kinds of flowers with six petals.
Figure
Three kinds of outlines of the vase.
Several existing basis functions containing just one shape parameter in [
We take [
In the previous work, the difference between the degree and order of the curve is fixed (i.e.,
However, comparing Figures
In this paper, a series of univariate Quasi-Bernstein basis functions are constructed, thereby creating a series of Quasi-Bézier curves. The shape of the series of curves can be adjusted even with the control points fixed. The Quasi-Bézier curves also possess geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints, and tangent edges at the endpoints.
Quasi-Bernstein basis functions with shape parameters have been directly studied in the previous research. However, in this paper, each function has been gradually inferred and constructed using a clear method of undetermined coefficients, where each shape parameter is proposed according to the properties of the Quasi-Bernstein basis functions and the Quasi-Bézier curve. Under the premise of satisfying symmetry, the former basis functions are all considered as the special cases in this paper.
In the existing CAD/CAM systems, the triangular Bézier surface and the spline curve are widely used. The shape parameters also have been brought into the triangular surface in [
The author is very grateful to the anonymous referees for the inspiring comments and the valuable suggestions which improved the paper considerably. This work has been supported by the National Natural Science Foundation of China (no. Y5090377) and the Natural Science Foundation of Ningbo (nos. 2011A610174 and 2012A610029).