Quasi-Bézier Curves with Shape Parameters

The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding QuasiBézier curveswere constructed step-by-step in this paper, using themethod of undetermined coefficients.The series ofQuasi-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.


Introduction
The Bézier curve  1 () listed as follows has a direct-viewing structure and can be computed using a simple process; it is also one of the most important tools in computer-aided geometric design (CAGD).Consider Here, Bernstein basis functions {   ()}  =0 are defined as: Given that the shape of the curve is characterized by the control polygon, the designer always adjusts the control point {P  }  =0 when necessary.However, in the actual process, designing the geometrical shape is usually not completed at one time.The designer prefers to have more satisfactory geometrical shapes by maintaining control polygon, which allows him or her to make minute adjustments on the shape of the curve with fixed control points.
The rational Bézier curve  2 () listed as follows is a natural choice to meet this requirement [1].
By assigning a weight   for each control point P  , the designer can adjust the shape of the curve by changing the value of the weights {  }  =0 [2,3].Although the rational Bézier curve has good properties and can express the conic section, it 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.
In addition, the algebraic trigonometric/hyperbolic curve  3 () with the definition domain  as the shape parameter is a feasible method [4][5][6].Consider The simple form of the algebraic trigonometric/hyperbolic curve  3 () can express transcendental curves (e.g., spiral and cycloid) that cannot be expressed by the Bézier curve.Nevertheless, the basis functions {   ()}  =0 include trigonometric/hyperbolic functions, such as sin , cos , sinh , and cosh .So, the algebraic trigonometric/hyperbolic curve is incompatible with the existing NURBS system, thereby restricting its application in the actual project.
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 Curve with multiple shape parameters Geometric and affine invariability The property of symmetry in [8][9][10] is based on the shape parameters.
The construction of the basis functions with shape parameters is the key step in [7][8][9][10][11][12].Although many kinds of basis functions with shape parameters have been obtained in the existing research, two problems need to be solved.
(1) 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?
(2) 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.

Quasi-Bézier Curve
2.1.Notation.First, the following vectors are introduced: Equation ( 5) can be rewritten as Given that { =0 are polynomials with degree  2 , they can be seen as the linear combination of the Bernstein basis functions { Thus, as long as the elements in the matrix M  2 , 1 are determined, the Quasi-Bernstein basis functions { with order  1 and degree  2 are completely constructed.Except for several elements that can be determined in M  2 , 1 , the rest are shape parameters of the Quasi-Bernstein basis functions and the Quasi-Bézier curve.Here, the matrix M  2 , 1 is called the shape parameter matrix.

Construction of the Shape Parameter Matrix
=0 and P() become the Quasi-Bernstein basis functions and the Quasi-Bézier curve, respectively.

Determination of 𝑚 𝑖𝑗 according to the Characteristics of the Quasi-Bernstein Basis Functions. The Quasi-Bernstein basis functions {𝑏
=0 with order  1 and degree  2 must satisfy the characteristics of nonnegativity, normalization, symmetry, linear independence, and degeneracy.

Proposition 1 (nonnegativity). A sufficient condition for 𝑏
Proof.Here, () is known to have been extracted from (8).Based on the non-negativity of the Bernstein basis functions { =0 , a sufficient condition for the non-negativity of the Quasi-Bernstein basis functions { =0 is the non-negativity of the elements   in M  2 , 1 .Hence,   must satisfy (9).Note 1.Clearly, there is no row with all elements being 0 in M  2 , 1 .In other words,

Proposition 4 (linear independence). The necessary and sufficient condition for the linear independence of {𝑏
is given by Proof.It is known that According to the linear independence of the Bernstein basis functions { =0 , the necessary and sufficient condition for ∑ M  = ( 0, ,  1, , . . .,   2 , )  is defined as the th column vector of M  2 , 1 .Equation ( 17) is equivalent to Thus, the necessary and sufficient condition for the linear independence of { =0 is also the linear independence of the column vectors {M  }  1 =0 of the matrix M  2 , 1 .Consequently, the necessary and sufficient condition for the linear independence of { =0 is that the rank of the shape parameter matrix M  2 , 1 satisfies (M  2 , 1 ) =  1 + 1.
, 1 is an identity matrix here.
Similarly, the necessary and sufficient condition for =0 have the property of symmetry, ( 27) is equivalent to (26) according to (13).
(b) 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; (c) affine invariability: barycentric combinations are invariant under affine maps; therefore, ( 9) and (11) give the algebraic verification of this property; (d) symmetry: whether the control points are labeled P 0 P 1 ⋅ ⋅ ⋅ P  1 or P  1 P  1 −1 ⋅ ⋅ ⋅ P 0 , the curves that correspond to the two different orderings look the same; they differ only in the direction in which they are traversed, and this is written as which follows the inspection of ( 13); (e) convex hull property: this property exists since the Quasi-Bernstein basis functions { =0 have the properties of non-negativity and normalization; the Quasi-Bézier curve is the convex linear combination of control points, and as such, it is located in the convex hull of the control points; (f) interpolation at the endpoints and tangent edges at the endpoint: the Quasi-Bézier curve P() interpolates the first and the last control points P(0) = P 0 and P(1) = P  1 ; the first and last edges of the control polygon are the tangent lines at the endpoints, where P  (0)‖P 1 P 0 and P  (1)‖P  P −1 .

Numerical Examples
Example 1.The shape parameter matrix M 3,2 is constructed from (29).The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter  10 are given as follows: The geometric significance of the shape parameters  10 is shown in Figure 1.As the value of the shape parameter  10 increases, the elements in the second column of M 3,2  decrease.According to (8), the second Quasi-Bernstein basis function  2,3 1 () decreases.So, the corresponding Quasi-Bézier curve moves away from the control point P 1 (see Figure 1(b)).(33) The geometric significance of the shape parameters  10 and  20 is shown in Figure 2. When we increase the value of  10 and keep  20 unchanged, the elements in the second column of M 4,2 decrease.According to (8), the second Quasi-Bernstein basis function  2,4  1 () decreases.Compare the blue curve with the red one, and we will find that the Quasi-Bézier curve moves away from the control point P 1 (see Figure 2(b)).
If we increase the value of  20 and keep  10 unchanged, similar result is also obtained.Compare the red curve with the green one, and we will find that the Quasi-Bézier curve moves away from the control point P 1 (see Figure 2(b)).
Example 3. The shape parameter matrix M 3,3 is constructed from (29).The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter  10 are given as follows: The geometric significance of the shape parameters  10 is shown in Figure 3.As the value of the shape parameter  10 increases, the elements in the second and the third column of M 3,3 decrease.According to (8), the second Quasi-Bernstein basis function  3,3  1 () and the third Quasi-Bernstein basis function  3,3  2 () decrease.So, the corresponding Quasi-Bézier curve moves away from the control points P 1 and P 2 (see Figure 3(b)).functions and the Quasi-Bézier curves with different shape parameter  10 and  20 are given as follows: The geometric significance of the shape parameters  10 and  20 is shown in Figure 4.When we increase the value of  10 and keep  20 unchanged, the elements in the second and the third column of M 4,3 decrease.According to (8), the second Quasi-Bernstein basis function  3,4  1 () and the third Quasi-Bernstein basis function  3,4  2 () decrease.Compare the blue curve with the red one, and we will find that the Quasi-Bézier curve moves away from the control points P 1 and P 2 (see Figure 4(b)).
If we increase the value of  20 and keep  10 unchanged, similar result is also obtained.Compare the red curve with the green one, and we will find that the Quasi-Bézier curve moves away from the control points P 1 and P 2 (see Figure 4   Note 7. 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 [7,12]:  designer can make minute adjustments with the same control polygons by changing the value of the shape parameters.

Discussion
4.1.Special Cases.Several existing basis functions containing just one shape parameter in [7,12] are considered as the special cases in this paper.Meanwhile, for the polynomial basis functions with multiple shape parameters in [8][9][10][11], the symmetry was not discussed by authors.In fact, when these shape parameters satisfy certain relations, the corresponding basis functions and curves become symmetrical.Then, the curves have geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints, and

Figure 5 :
Figure 5: Three kinds of flowers with six petals.

2 − 1 ( 2 𝑛 1
) =   1 , +1 () = 0. Example 5. Figure 5 presents three kinds of flowers with six petals, defined by six symmetric control polygons.Similar flowers are obtained from the same control polygons with different shape parameters.Example 6. Figure 6 presents three kinds of outlines of the vase, all of which are similar to the control polygons.So, the

Figure 6 :
Figure 6: Three kinds of outlines of the vase.

Table 1 :
Properties of the basis functions and the curves with shape parameters.