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  浙江大学学报(理学版)  2018, Vol. 45 Issue (1): 44-53  DOI:10.3785/j.issn.1008-9497.2018.01.008
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YAN Lanlan, FAN Jiqiu. Construction and analysis of a new class of shape-preserving piecewise cubic polynomial curves[J]. Journal of Zhejiang University(Science Edition), 2018, 45(1): 44-53. DOI: 10.3785/j.issn.1008-9497.2018.01.008.
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严兰兰, 樊继秋. 保形分段三次多项式曲线的形状分析[J]. 浙江大学学报(理学版), 2018, 45(1): 44-53. DOI: 10.3785/j.issn.1008-9497.2018.01.008.
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Fundation item

Supported by the NSFC(11261003, 11761008), the Natural Science Foundation of Jiangxi Province (20161BAB211028) and the Science Research Foundation of Jiangxi Province Education Department (GJJ160558)

About the author

YAN Lanlan(1982-), ORCID:http://orcid.org/0000-0002-5472-9986, female, Ph.D, associate professor, the field of interest is CAGD, E-mail:yxh821011@aliyun.com

Article History

Received Date: April 5, 2016
Construction and analysis of a new class of shape-preserving piecewise cubic polynomial curves
YAN Lanlan , FAN Jiqiu     
College of Science, East China University of Technology, Nanchang 330013, China
Received Date: April 5, 2016
Fundation item: Supported by the NSFC(11261003, 11761008), the Natural Science Foundation of Jiangxi Province (20161BAB211028) and the Science Research Foundation of Jiangxi Province Education Department (GJJ160558)
About the author: YAN Lanlan(1982-), ORCID:http://orcid.org/0000-0002-5472-9986, female, Ph.D, associate professor, the field of interest is CAGD, E-mail:yxh821011@aliyun.com
Abstract: This paper proposes a new class of shape-preserving piecewise cubic polynomial curves with both local and global shape control parameters. By presetting the properties of its basis functions and then solving equations, a set of polynomial basis functions with two shape parameters are derived, including the cubic uniform B-spline basis functions as a special case. Based on the relationship between the new basis functions and the cubic Bernstein basis functions, the totally positive property of the new basis functions is proved and a new class of piecewise cubic polynomial curves is therefore defined. The effect of the relative position of the control polygons' side vectors onto the shape characteristic of the corresponding curve segments is analyzed. Necessary and sufficient conditions are obtained for the curve segments containing single or double inflection points, a loop or a cusp, or be locally or globally convex, which provide a theoretical guide for adjusting the shape of curve segments.
Key words: curve design    B-spline method    totally positive basis    shape parameter    shape analysis    
保形分段三次多项式曲线的形状分析
严兰兰, 樊继秋    
东华理工大学 理学院, 江西 南昌 330013
摘要: 构造了一种保形并且形状可调的分段三次多项式曲线,并分析其形状特征与控制多边形之间的关系.首先,通过预设基函数的性质再解方程组,构造了一组带2个形状参数的多项式基函数,其包含三次均匀B样条基函数作为特例.然后,借助基函数与三次Bernstein基函数之间的关系证明了基函数的全正性,由这组基函数定义了一种分段三次多项式曲线,使该曲线拥有一个局部和一个全局形状参数.最后,分析了控制多边形边变量之间的相对位置关系对曲线段形状特征的影响,得到了曲线段拥有1个或2个拐点,1个二重点或1个尖点,为局部凸或全局凸时的充要条件.该结论为曲线段的形状调整提供了理论基础.
关键词: 曲线设计    B样条方法    全正基    形状参数    形状分析    

Constructing practical basis functions to generate free-form curves and surfaces is an important topic in computer aided geometric design (CAGD).As a unified mathematical model with many desirable properties, B-splines, particularly the cubic B-splines, have gained widespread application in CAGD [1].Piecewise cubic B-spline curves with four consecutive control points for each curve segment is flexible and can be used conveniently.However, the positions of the cubic B-spline curves are fixed relatively to their control polygon.Although the weights in the cubic non-uniform rational B-spline curves possess an effect on adjusting the shape of the curves, how to change the weights to adjust the shape of a curve is sometimes quite opaque to the user.

To enhance the flexibility of B-spline models, some researchers have suggested many types of curves with shape parameters incorporated into the basis functions.For instance, XU et al[2] proposed three kinds of extensions of cubic uniform B-spline. The advantage of the extensions is that they have shape parameters, which can be used to adjust the shape of the curves without shifting the control points.COSTANTINI et al[3] presented a method for the construction of cubic like B-splines with multiple knots.The proposed B-splines are equipped with tension parameters, associated to the knots, which permit a modification of their shape.HAN[4] constructed piecewise quartic polynomial curves with a local shape parameter.HAN[5] defined piecewise quartic spline curves with three local shape parameters.HU et al[6] presented B-spline curves with two local shape parameters.ZHU et al[7] defined B-spline-like curves with two local shape parameters.

Totally positive property is one of the most important properties of basis functions.Although the schemes given in [2, 4-6] improved the control of the shape of B-spline curves, whether the basis functions have total positivity is unknown, so whether the curves have variation diminishing is unknown.Although the curves in [3, 7] have variation diminishing, the basis functions are not cubic polynomials.For many applications in geometric modeling, it is often necessary to detect singularities and inflection points on curves, and convexity is important as an intuitive geometric concept as well.There are many publications[8-13] on this topic from different points of view.

This paper is aimed at constructing a cubic polynomial basis functions with total positivity.The associated curves have local control, adjustable shape, and have variation diminishing thus have a good shape control.Considering that a curve with variation diminishing is suitable for conformal design, we analyze the shape feature of the new curves.We give conditions on the existence of cusp, loop and inflection point.The results are summarized in a shape diagram like the one in [8, 10-11].Furthermore, the influence of the shape parameters on the shape diagram and their ability for adjusting the shape of the curves is discussed.

The rest of the paper is organized as follows.Section 1 gives the basis functions and their properties.Section 2 defines the curves and gives their properties.Section 3 analyzes the shape feature of the curve segments.Section 4 defines the surfaces.Section 5 concludes the paper.

1 Basis functions and their properties

Definition 1 For t∈[0, 1], $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0], the following four functions are defined to be the cubic polynomial basis functions with parameters α and β (αβ basis for short) :

$ \left\{ \begin{array}{l} {b_0}\left( {t;\alpha ,\beta } \right) = \frac{{\alpha - 3\beta }}{6}{t^3} - \frac{{\alpha - 2\beta }}{2}{t^2} + \frac{{\alpha - \beta }}{2}t - \frac{\alpha }{6},\\ {b_1}\left( {t;\alpha ,\beta } \right) = \frac{{4 + 3\alpha - \beta }}{2}{t^3} - \frac{{6 + 4\alpha - \beta }}{2}{t^2} + \frac{\alpha }{3} + 1,\\ {b_2}\left( {t;\alpha ,\beta } \right) = - \frac{{4 + 3\alpha - \beta }}{2}{t^3} + \frac{{6 + 5\alpha - 2\beta }}{2}{t^2} - \frac{{\alpha - \beta }}{2}t - \frac{\alpha }{6},\\ {b_3}\left( {t;\alpha ,\beta } \right) = - \frac{{\alpha - 3\beta }}{6}{t^3} - \frac{\beta }{2}{t^2}. \end{array} \right. $ (1)

For brevity, we will denote αβ basis as {bi(t) }i=03 whenever there is confusion or not.

The αβ basis can be rewritten in Bernstein basis functions form, i.e.,

$ \begin{array}{l} \left( {{b_0}\left( t \right),{b_1}\left( t \right),{b_2}\left( t \right),{b_3}\left( t \right)} \right) = \\ \;\;\;\;\;\;\left( {B_0^3\left( t \right),B_1^3\left( t \right),B_2^3\left( t \right),B_3^3\left( t \right)} \right)\mathit{\boldsymbol{J}}, \end{array} $ (2)

where {Bi3(t) }i=03 are the cubic Bernstein basis functions, and

$ \mathit{\boldsymbol{J}} = \left( {\begin{array}{*{20}{c}} { - \frac{\alpha }{6}}&{1 + \frac{\alpha }{3}}&{ - \frac{\alpha }{6}}&0\\ { - \frac{\beta }{6}}&{1 + \frac{\alpha }{3}}&{\frac{\beta }{6} - \frac{\alpha }{3}}&0\\ 0&{\frac{\beta }{6} - \frac{\alpha }{3}}&{1 + \frac{\alpha }{3}}&{ - \frac{\beta }{6}}\\ 0&{ - \frac{\alpha }{6}}&{1 + \frac{\alpha }{3}}&{ - \frac{\alpha }{6}} \end{array}} \right). $

Lemma 1 When $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0], J is a nonsingular stochastic and totally positive matrix.

Proof When $ \alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0], all elements of the matrix J are non-negative and the sum of each row elements is equal to 1.These means that J is stochastic.The matrix J has 36 second-order minors as follows:

$ \left\{ \begin{array}{l} \left| {{\mathit{\boldsymbol{J}}_{12,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{12,23}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{34,23}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{34,34}}} \right| = \frac{{\left( {3 + \alpha } \right)\left( {\beta - \alpha } \right)}}{{18}},\\ \left| {{\mathit{\boldsymbol{J}}_{13,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{24,34}}} \right| = \frac{{ - \alpha \left( {\beta - 2\alpha } \right)}}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{24,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{13,14}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{24,14}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{13,34}}} \right| = \frac{{\alpha \beta }}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{23,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{23,34}}} \right| = \frac{{ - \beta \left( {\beta - 2\alpha } \right)}}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{13,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{14,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{14,24}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{24,24}}} \right| = \frac{{ - \alpha \left( {3 + \alpha } \right)}}{{18}},\\ \left| {{\mathit{\boldsymbol{J}}_{12,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{34,24}}} \right| = \frac{{ - \alpha \left( {\beta - \alpha } \right)}}{{18}},\\ \left| {{\mathit{\boldsymbol{J}}_{23,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{24,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{13,24}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{23,24}}} \right| = \frac{{ - \beta \left( {3 + \alpha } \right)}}{{18}},\\ \left| {{\mathit{\boldsymbol{J}}_{13,23}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{24,23}}} \right| = \frac{{36 + 24\alpha + 2{\alpha ^2} + \alpha \beta }}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{14,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{14,14}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{14,34}}} \right| = \frac{{{\alpha ^2}}}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{23,14}}} \right| = \frac{{{\beta ^2}}}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{23,23}}} \right| = \frac{{\left( {6 + 4\alpha - \beta } \right)\left( {6 + \beta } \right)}}{{36}},\\ \left| {{\mathit{\boldsymbol{J}}_{14,23}}} \right| = \frac{{\left( {2 + \alpha } \right)\left( {6 + \alpha } \right)}}{{12}},\\ \left| {{\mathit{\boldsymbol{J}}_{34,12}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{34,13}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{34,14}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{12,14}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{12,24}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{12,34}}} \right| = 0, \end{array} \right. $

where |Jij, kl| denotes the minor formed by the i, j rows and k, l columns of J.The matrix J has 16 third-order minors as follows:

$ \left\{ \begin{array}{l} \left| {{\mathit{\boldsymbol{J}}_{123,123}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{234,234}}} \right| = \frac{{\left( {18 + 12\alpha + \alpha \beta } \right)\left( {\beta - \alpha } \right)}}{{108}},\\ \left| {{\mathit{\boldsymbol{J}}_{124,134}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{134,124}}} \right| = \frac{{{\alpha ^2}\left( {\beta - \alpha } \right)}}{{108}},\\ \left| {{\mathit{\boldsymbol{J}}_{134,123}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{124,124}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{134,134}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{124,234}}} \right| = \frac{{ - \alpha \left( {3 + \alpha } \right)\left( {\beta - \alpha } \right)}}{{108}},\\ \left| {{\mathit{\boldsymbol{J}}_{234,123}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{123,124}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{234,134}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{123,234}}} \right| = \frac{{ - \beta \left( {3 + \alpha } \right)\left( {\beta - \alpha } \right)}}{{108}},\\ \left| {{\mathit{\boldsymbol{J}}_{124,123}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{134,234}}} \right| = \frac{{\left( {18 + 12\alpha + {\alpha ^2}} \right)\left( {\beta - \alpha } \right)}}{{108}},\\ \left| {{\mathit{\boldsymbol{J}}_{234,124}}} \right| = \left| {{\mathit{\boldsymbol{J}}_{123,134}}} \right| = \frac{{\alpha \beta \left( {\beta - \alpha } \right)}}{{108}}, \end{array} \right. $

where |Jijk, lmn| denotes the minor formed by the i, j, k rows and l, m, n columns of J.Besides,

$ \left| \mathit{\boldsymbol{J}} \right| = \frac{{\left( {3 + 2\alpha } \right){{\left( {\beta - \alpha } \right)}^2}}}{{108}}. $ (3)

When $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0], it is easy to judge that all the minors of J are nonnegative.Therefore, J is a nonsingular stochastic and totally positive matrix.

Theorem 1 The αβ basis has the following properties:

(a) Degeneracy: When α=-1, β=0, the αβ basis is the cubic uniform B-spline basis.

(b) Partition of unity:$\sum\limits_{i = 0}^3 {{b_i}\left( t \right) \equiv 1} $.

(c) Symmetry: bi(1-t) =b3-i(t) (i=0, 1, 2, 3).

(d) Property at the endpoints: For arbitrary α and β, we have

$ \left\{ \begin{array}{l} {b_0}\left( 0 \right) = {b_1}\left( 1 \right) = {b_2}\left( 0 \right) = {b_3}\left( 1 \right) = - \frac{\alpha }{6},\\ {b_1}\left( 0 \right) = {b_2}\left( 1 \right) = 1 + \frac{\alpha }{3},{b_3}\left( 0 \right) = {b_0}\left( 1 \right) = 0,\\ {{b'}_0}\left( 0 \right) = {{b'}_1}\left( 1 \right) = \frac{{\alpha - \beta }}{2},{{b'}_2}\left( 0 \right) = {{b'}_3}\left( 1 \right) = \frac{{\beta - \alpha }}{2},\\ {{b'}_1}\left( 0 \right) = {{b'}_2}\left( 1 \right) = {{b'}_3}\left( 0 \right) = {{b'}_0}\left( 1 \right) = 0, \end{array} \right. $

and for α=-1, β=0, we have

$ \left\{ \begin{array}{l} {{b''}_0}\left( 0 \right) = {{b''}_1}\left( 1 \right) = {{b''}_2}\left( 0 \right) = {{b''}_3}\left( 1 \right) = 1,\\ {{b''}_1}\left( 0 \right) = {{b''}_2}\left( 1 \right) = - 2,{{b''}_3}\left( 0 \right) = {{b''}_0}\left( 1 \right) = 0. \end{array} \right. $

(e) Linear independence: The set {bi(t) }i=03 is linearly independent on [0, 1].

(f) Non-negativity: bi(t) ≥0(i=0, 1, 2, 3).

(g) Total positivity: The αβ basis forms a normalized totally positive basis of the space Ω={1, t, t2, t3}.

Proof We shall prove (e) and(g).The remaining can be easily obtained by formula (1) or (2).

(e) By (3), when $\alpha \ne-\frac{3}{2} $, βα, we have |J|≠0. Notice that the cubic Bernstein basis is linearly independent, hence by (2), we know the αβ basis is also linearly independent.

(g) The cubic Bernstein basis is the normalized B-basis of the space Ω. Thus, by formula (2), lemma 1, and the properties (b) and (e), we know the αβ basis is a normalized totally positive basis.

2 The αβ curves and their properties

Definition 2 Given control points Pi(i=0, 1, …, n) ∈R2 or R3 and knots u1 < u2 < … < un-1, then we can define (n-2) th αβ curve segments as follows:

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{p}}_i}\left( t \right) = \sum\limits_{j = 0}^3 {{\mathit{\boldsymbol{P}}_{i + j - 1}}{b_j}\left( {t;\alpha ,{\beta _i}} \right)} ,t \in \left[ {0,1} \right],}\\ {t = 1,2, \cdots ,n - 2,} \end{array} $

where {bj(t; α, βi) }j=03 is αβ basis.All the segments constitute an αβ curve

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{q}}\left( u \right) = {\mathit{\boldsymbol{p}}_i}\left( {\frac{{u - {u_i}}}{{{u_{i + 1}} - {u_i}}}} \right),u \in \left[ {{u_i},{u_{i + 1}}} \right],}\\ {t = 1,2, \cdots ,n - 2.} \end{array} $

Theorem 2 From the properties of the αβ basis, we can obtain the following properties of the αβ curves.

(a) Affine invariance.

(b) Symmetry: When taking the same αβ basis, the two polygons, Pi-1, Pi, Pi+1, Pi+2 and Pi+2, Pi+1, Pi, Pi-1, describe the same αβ curve segment; The only thing that changes is the direction of traversal of the parameter.

(c) Local control: Changing one control point of an αβ curve, four segments will change mostly.

(d) Endpoint property: For arbitrary α and βi, we have

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{p}}_i}\left( 0 \right) = - \frac{\alpha }{6}{\mathit{\boldsymbol{P}}_{i - 1}} + \left( {1 + \frac{\alpha }{3}} \right){\mathit{\boldsymbol{P}}_i} - \frac{\alpha }{6}{\mathit{\boldsymbol{P}}_{i + 1}},\\ {{\mathit{\boldsymbol{p'}}}_i}\left( 0 \right) = \frac{{{\beta _i} - \alpha }}{2}\left( {{\mathit{\boldsymbol{P}}_{i + 1}} - {\mathit{\boldsymbol{P}}_{i - 1}}} \right),\\ {\mathit{\boldsymbol{p}}_i}\left( 1 \right) = - \frac{\alpha }{6}{\mathit{\boldsymbol{P}}_i} + \left( {1 + \frac{\alpha }{3}} \right){\mathit{\boldsymbol{P}}_{i + 1}} - \frac{\alpha }{6}{\mathit{\boldsymbol{P}}_{i + 2}},\\ {{\mathit{\boldsymbol{p'}}}_i}\left( 1 \right) = \frac{{{\beta _i} - \alpha }}{2}\left( {{\mathit{\boldsymbol{P}}_{i + 2}} - {\mathit{\boldsymbol{P}}_i}} \right), \end{array} \right. $

and for α=-1, βi=0, we have

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{p''}}}_i}\left( 0 \right) = {\mathit{\boldsymbol{P}}_{i - 1}} - 2{\mathit{\boldsymbol{P}}_i} + {\mathit{\boldsymbol{P}}_{i + 1}},\\ {{\mathit{\boldsymbol{p''}}}_i}\left( 1 \right) = {\mathit{\boldsymbol{P}}_i} - 2{\mathit{\boldsymbol{P}}_{i + 1}} + {\mathit{\boldsymbol{P}}_{i + 2}}. \end{array} \right. $

Remark By the endpoint property, we can see that the position of the starting and ending points of the curve segment only related to the parameter α. The value βiα decides to the length of the tangent vector at the starting and ending points, thus decides to the degree of blending of the ith curve segment.

(e) Continuity: In general, the i th and (i+1) th αβ curve segments are G1 continuous at the junction, and they are G2 continuous when α=-1, βi=βi+1=0.

(f) Shape adjustable: The parameters α and βi can be used to adjust the shape of the αβ curve without changing the control points.The α is a global parameter, while βi is a local parameter.The change of βi will only change the shape of the ith segment.

(g) Convex hull: The αβ curve segment pi(t) lies inside the convex hull Hi of the control points Pi-1, Pi, Pi+1, Pi+2, and the entire curve q(u) lies inside $H = \bigcup\limits_{i = 1}^{n-2} {{H_i}} $, which is the union of Hi.

(h) Variation diminishing and convexity-preserving.

Figs. 1 and 2 show the αβ curves defined by the same control points but different parameters.In fig. 1(a) and fig. 2(a), α=-1, all ${\beta _i} =-\frac{1}{2} $. In fig. 1(b) and fig. 2(b), $\alpha =-\frac{1}{2} $, all βi=0.

Fig. 1 The open αβ curves
Fig. 2 The closed αβ curves

As can be seen in figs. 1 and 2, the shape of αβ curves can reflect the shape of the control polygon well.

3 Shape analysis of the curve segment

We consider an αβ curve segment

$ \mathit{\boldsymbol{p}}\left( t \right) = \sum\limits_{i = 0}^3 {{\mathit{\boldsymbol{P}}_i}{b_i}\left( {t;\alpha ,\beta } \right)} , $

where t∈[0, 1], $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0]. Denote

$ {\mathit{\boldsymbol{a}}_i} = {\mathit{\boldsymbol{P}}_i} - {\mathit{\boldsymbol{P}}_{i - 1}},i = 1,2,3, $

then p(t) can be rewritten as

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{p}}\left( t \right) = {\mathit{\boldsymbol{P}}_0} + \left[ {1 - {b_0}\left( t \right)} \right]{\mathit{\boldsymbol{a}}_1} + \left[ {{b_2}\left( t \right) + } \right.}\\ {\left. {{b_3}\left( t \right)} \right]{\mathit{\boldsymbol{a}}_2} + {b_3}\left( t \right){\mathit{\boldsymbol{a}}_3}.} \end{array} $ (4)

We first consider the case of a1 not parallel to a3. Since a1 and a3 are linearly independent, a2 can be represented by the linear combination of a1 and a3.Without loss of generality, let a2=ua1+va3, and substitute it into formula (4), and then we have

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{p}}\left( t \right) = {\mathit{\boldsymbol{P}}_0} + \left\{ {1 - {b_0}\left( t \right) + u\left[ {{b_2}\left( t \right) + {b_3}\left( t \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_1} + }\\ {\left\{ {{b_3}\left( t \right) + v\left[ {{b_2}\left( t \right) + {b_3}\left( t \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_3}.} \end{array} $ (5)
3.1 The case of cusp

The necessary condition that the curve p(t) has a cusp is p′(t) =0(0 < t < 1). According to formula (5),

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{p'}}\left( t \right) = \left\{ { - {{b'}_0}\left( t \right) + u\left[ {{{b'}_2}\left( t \right) + {{b'}_3}\left( t \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_1} + }\\ {\left\{ {{{b'}_3}\left( t \right) + v\left[ {{{b'}_2}\left( t \right) + {{b'}_3}\left( t \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_3}.} \end{array} $

Let p′(t) =0, for a1 and a3 are linearly independent, we obtain

$ - {{b'}_0}\left( t \right) + u\left[ {{{b'}_2}\left( t \right) + {{b'}_3}\left( t \right)} \right] = 0, $
$ {{b'}_3}\left( t \right) + v\left[ {{{b'}_2}\left( t \right) + {{b'}_3}\left( t \right)} \right] = 0, $

and then the following parametric curve $C \buildrel \Delta \over = C\left( t \right) $ can be obtained.

$ C:\left\{ \begin{array}{l} u\left( t \right) = \frac{{b{{_0^3}^\prime }\left( t \right)}}{{b{{_2^3}^\prime }\left( t \right) + b{{_3^3}^\prime }\left( t \right)}} = \\ \;\;\;\;\;\;\;\;\;\;\frac{{\left( {1 - t} \right)\left[ {\left( {3\beta - \alpha } \right)t + \left( {\alpha - \beta } \right)} \right]}}{{2\left( {6 + 5\alpha - 3\beta } \right)t\left( {1 - t} \right) + \left( {\beta - \alpha } \right)}},\\ v\left( t \right) = - \frac{{b{{_3^3}^\prime }\left( t \right)}}{{b{{_2^3}^\prime }\left( t \right) + b{{_3^3}^\prime }\left( t \right)}} = \\ \;\;\;\;\;\;\;\;\;\;\frac{{t\left[ {\left( {\alpha - 3\beta } \right)t + 2\beta } \right]}}{{2\left( {6 + 5\alpha - 3\beta } \right)t\left( {1 - t} \right) + \left( {\beta - \alpha } \right)}}, \end{array} \right.t \in \left( {0,1} \right). $ (6)

To facilitate the analysis of the geometric properties of C, we rewrite it in the form of quadratic rational Bézier curve, i.e,

$ C\left( t \right) = \frac{{\sum\limits_{i = 0}^2 {{\omega _i}{Q_i}{B_i}\left( t \right)} }}{{\sum\limits_{i = 0}^2 {{\omega _i}{B_i}\left( t \right)} }}, $

where Bi(t) =C2iti(1-t) 2-i(i=0, 1, 2), the control points Qi(i=0, 1, 2) (two-dimensional column vector in the uv-plane) and weights ωi(i=0, 1, 2) are as follows

$ \left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{Q}}_0}}&{{\mathit{\boldsymbol{Q}}_1}}&{{\mathit{\boldsymbol{Q}}_2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 1}&{\frac{\beta }{{2\left( {3 + 2\alpha - \beta } \right)}}}&0\\ 0&{\frac{\beta }{{2\left( {3 + 2\alpha - \beta } \right)}}}&{ - 1} \end{array}} \right), $ (7)
$ \left( {\begin{array}{*{20}{c}} {{\omega _0}}&{{\omega _1}}&{{\omega _2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\beta - \alpha }&{2\left( {3 + 2\alpha - \beta } \right)}&{\beta - \alpha } \end{array}} \right). $ (8)

When $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, 0, β∈(α, 0], the control polygon given by formula (7) is global convex, and the weights given by formula (8) are positive.Hence, C is a global convex curve.In addition, C is symmetric about the straight line u=v. When t=0, the curve C and u-axis intersect at (-1, 0). When t=1, the curve C and v-axis intersect at (0, -1). Fig. 3 shows these characteristics.

Fig. 3 The shape diagram of the αβ curve

Let (u0, v0) ∈C, and 0 < t0 < 1 is the corresponding parameter.If p″(t0) =0, then β=α can be deduced.But here β>α, so p″(t0) ≠0. Then from

$ \mathit{\boldsymbol{p'}}\left( t \right) = \mathit{\boldsymbol{p''}}\left( {{t_0}} \right)\left( {t - {t_0}} \right) + O\left( {t - {t_0}} \right), $

we can see that the direction of p′(t) is contravariant when it passes through t0.It shows that C is the cusp curve, that is, the curve p(t) has cusp if and only if (u, v) ∈C.

3.2 The case of loop

The sufficient and necessary condition that the curve p(t) has a loop is that there exists 0≤t1 < t2≤1 so that p(t1) =p(t2).According to formula(5), it is equivalent to

$ \begin{array}{l} \left\{ {{b_0}\left( {{t_2}} \right) - {b_0}\left( {{t_1}} \right) + u\left[ {{b_2}\left( {{t_1}} \right) + {b_3}\left( {{t_1}} \right) - {b_2}\left( {{t_2}} \right) - } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {{b_3}\left( {{t_2}} \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_1} + \left\{ {{b_3}\left( {{t_1}} \right) - {b_3}\left( {{t_2}} \right) + v\left[ {{b_2}\left( {{t_1}} \right) + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {{b_3}\left( {{t_1}} \right) - {b_2}\left( {{t_2}} \right) - {b_3}\left( {{t_2}} \right)} \right]} \right\}{\mathit{\boldsymbol{a}}_3} = 0. \end{array} $

For a1 and a3 are linearly independent, we obtain

$ \left\{ \begin{array}{l} u = \frac{{{b_0}\left( {{t_1}} \right) - {b_0}\left( {{t_2}} \right)}}{{{b_2}\left( {{t_1}} \right) + {b_3}\left( {{t_1}} \right) - {b_2}\left( {{t_2}} \right) - {b_3}\left( {{t_2}} \right)}},\\ v = \frac{{{b_3}\left( {{t_2}} \right) - {b_3}\left( {{t_1}} \right)}}{{{b_2}\left( {{t_1}} \right) + {b_3}\left( {{t_1}} \right) - {b_2}\left( {{t_2}} \right) - {b_3}\left( {{t_2}} \right)}}, \end{array} \right. $ (9)

where (t1, t2) ∈Δ={(t1, t2) ∈R2|0≤t1 < t2≤1}.

Eq.(9) defines a topological mapping F: Δ⊂R2F(Δ) ⊂R2. The image region L=F(Δ) is a simply connected region on uv-plane.Fig. 3 shows the region.The three boundary curves of L are corresponding to the three boundary lines t1=t2, t1=0 and t2=1 of Δ, i.e., the curve C does not belong to L, ${L_1} \buildrel \Delta \over = {L_1}\left( t \right) $ and ${L_2} \buildrel \Delta \over = {L_2}\left( t \right) $ belong to L, respectively.The curve p(t) corresponding to the point (u, v) in the region L has one and only one loop.

The parametric equations of L1 and L2 are as follows:

$ {L_1}:\left\{ \begin{array}{l} u\left( t \right) = \\ \;\;\;\;\;\frac{{\left( {3\beta - \alpha } \right){t^2} + 3\left( {\alpha - 2\beta } \right)t + 3\left( {\beta - \alpha } \right)}}{{2\left( {6 + 5\alpha - 3\beta } \right){t^2} - 3\left( {6 + 5\alpha - 3\beta } \right)t + 3\left( {\beta - \alpha } \right)}},\\ v\left( t \right) = \\ \;\;\;\;\;\frac{{\left( {3\beta - \alpha } \right){t^2} - 3\beta t}}{{2\left( {6 + 5\alpha - 3\beta } \right){t^2} - 3\left( {6 + 5\alpha - 3\beta } \right)t + 3\left( {\beta - \alpha } \right)}}, \end{array} \right.t \in \left( {0,1} \right), $ (10)
$ {L_2}:\left\{ \begin{array}{l} u\left( t \right) = \\ \;\;\;\;\;\frac{{ - \alpha {{\left( {1 - t} \right)}^2} - 3\beta t\left( {1 - t} \right)}}{{2\left( {6 + 5\alpha - 3\beta } \right){t^2} - \left( {6 + 5\alpha - 3\beta } \right)t - \left( {6 + 2\alpha } \right)}},\\ v\left( t \right) = \\ \;\;\;\;\;\frac{{\left( {3\beta - \alpha } \right){t^2} + \left( {2 + \alpha } \right)\left( {1 + t} \right)}}{{2\left( {6 + 5\alpha - 3\beta } \right){t^2} - \left( {6 + 5\alpha - 3\beta } \right)t - \left( {6 + 2\alpha } \right)}}, \end{array} \right.t \in \left( {0,1} \right). $ (11)

The curves L1 and L2 can be rewritten in the form of quadratic rational Bézier curves

$ {L_j}\left( t \right) = \frac{{\sum\limits_{i = 0}^2 {{\omega _{ji}}{\mathit{\boldsymbol{O}}_{ji}}{B_i}\left( t \right)} }}{{\sum\limits_{i = 0}^2 {{\omega _{ji}}{B_i}\left( t \right)} }},j = 1,2, $

where the control points Oji and weights ωji(j=1, 2; i=0, 1, 2) are as follows:

$ \left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{O}}_{10}}}&{{\mathit{\boldsymbol{O}}_{11}}}&{{\mathit{\boldsymbol{O}}_{12}}}\\ {{\mathit{\boldsymbol{O}}_{20}}}&{{\mathit{\boldsymbol{O}}_{21}}}&{{\mathit{\boldsymbol{O}}_{22}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 1}&{\frac{\alpha }{{6 + 3\alpha - \beta }}}&{\frac{\alpha }{{6 + 2\alpha }}}\\ 0&{\frac{\beta }{{6 + 3\alpha - \beta }}}&{\frac{\alpha }{{6 + 2\alpha }}}\\ {\frac{\alpha }{{6 + 2\alpha }}}&{\frac{\beta }{{6 + 3\alpha - \beta }}}&0\\ {\frac{\alpha }{{6 + 2\alpha }}}&{\frac{\alpha }{{6 + 3\alpha - \beta }}}&{ - 1} \end{array}} \right), $ (12)
$ \left( {\begin{array}{*{20}{c}} {{\omega _{10}}{\omega _{11}}{\omega _{12}}}\\ {{\omega _{20}}{\omega _{21}}{\omega _{22}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {3\left( {\beta - \alpha } \right)}&{\frac{{3\left( {6 + 3\alpha - \beta } \right)}}{2}}&{6 + 2\alpha }\\ {6 + 2\alpha }&{\frac{{3\left( {6 + 3\alpha - \beta } \right)}}{2}}&{3\left( {\beta - \alpha } \right)} \end{array}} \right). $ (13)

When $ \alpha \in \left( {-\frac{3}{2}, 0} \right)$, 0, β∈(α, 0], the control polygons of L1 and L2 determined by eq.(12) are global convex, and the weights given by eq.(13) are positive.Hence L1 and L2 are global convex curves.

The two curves L1 and L2 are symmetric about the straight line u=v, and their intersection is (u*, v*) = $\left( {\frac{\alpha }{{6 + 2\alpha }}, \frac{\alpha }{{6 + 2\alpha }}} \right) $. When t=0, L1 and u-axis intersect at (-1, 0). When t=1, L2 and v-axis intersect at (0, -1). The tangent line of L1 at (u*, v*), denoted by ${l_1} \buildrel \Delta \over = {\mathit{\boldsymbol{l}}_1}\left( t \right) $, passes through (0, -1). The tangent line of L2 at (u*, v*), denoted by ${l_2} \buildrel \Delta \over = {\mathit{\boldsymbol{l}}_2}\left( t \right) $, passes through (-1, 0).The parametric equations of l1 and l2 are as follows:

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{l}}_1}\left( t \right) = \left( {1 - t} \right){\mathit{\boldsymbol{O}}_{12}} + t{\mathit{\boldsymbol{O}}_{22}},\\ {\mathit{\boldsymbol{l}}_2}\left( t \right) = \left( {1 - t} \right){\mathit{\boldsymbol{O}}_{20}} + t{\mathit{\boldsymbol{O}}_{10}}, \end{array} \right. $ (14)

where t∈(0, 1). Eliminate the parameter t, we obtain

$ \left\{ \begin{array}{l} {l_1}:v = \frac{{u\left( {1 + {v^ * }} \right)}}{{{u^ * }}} - 1,\;\;\;{u^ * } < u < 0,\\ {l_2}:v = \frac{{{v^ * }\left( {1 + u} \right)}}{{1 + {u^ * }}},\;\;\;\;\; - 1 < u < {u^ * }. \end{array} \right. $
3.3 The case of inflection point

The binormal vector of p(t) is γ(t) =p′(t) ∧p″(t), where p′(t) ∧p″(t) is the wedge product of p′(t) and p″(t). If the direction of γ(t) is changed when it passes through t0, then p(t0) (0 < t0 < 1) is an inflection point.

The binormal vector γ(t) can be represented as γ(t) =f(t; u, v) (a1a3), where

$ \begin{array}{*{20}{c}} {f\left( {t;u,v} \right) = - \left| {\begin{array}{*{20}{c}} {b{{_0^3}^\prime }\left( t \right)}&{b{{_3^3}^\prime }\left( t \right)}\\ {b{{_0^3}^{\prime \prime }}\left( t \right)}&{b{{_3^3}^{\prime \prime }}\left( t \right)} \end{array}} \right| + u\left| {\begin{array}{*{20}{c}} {b{{_2^3}^\prime }\left( t \right)}&{b{{_3^3}^\prime }\left( t \right)}\\ {b{{_2^3}^{\prime \prime }}\left( t \right)}&{b{{_3^3}^{\prime \prime }}\left( t \right)} \end{array}} \right| + }\\ {v\left| {\begin{array}{*{20}{c}} {b{{_0^3}^\prime }\left( t \right)}&{b{{_1^3}^\prime }\left( t \right)}\\ {b{{_0^3}^{\prime \prime }}\left( t \right)}&{b{{_1^3}^{\prime \prime }}\left( t \right)} \end{array}} \right| \buildrel \Delta \over = A + uB + vC,} \end{array} $

with

$ \left\{ \begin{array}{l} A = \frac{{\beta - \alpha }}{2}\left[ {\left( {3\beta - \alpha } \right)t\left( {1 - t} \right) - \beta } \right],\\ B = \frac{{\beta - \alpha }}{2}\left[ {\left( {6 + 5\alpha - 3\beta } \right){t^2} - \left( {\alpha - 3\beta } \right)t - \beta } \right],\\ C = \frac{{\beta - \alpha }}{2}\left[ {\left( {6 + 5\alpha - 3\beta } \right){t^2} - 3\left( {4 + 3\alpha - \beta } \right)t + } \right.\\ \;\;\;\;\;\left. {\left( {6 + 4\alpha - \beta } \right)} \right]. \end{array} \right. $

Since a1×a3≠0, the direction of γ(t) changes if and only if the sign of f(t; u, v) changes.Hence, we only need to consider the sign change of f(t; u, v).

On the uv-plane, the possible region that the curve p(t) has an inflection point must be covered by the family of straight lines f(t; u, v) =0.The envelope of f(t; u, v) =0 can be obtained by solving the equation of u and v: f(t; u, v) =0, ft(t; u, v) =0, and it is just the curve C (see eq.(6)).Since C is a strictly convex continuous curve, the region swept by the tangent line of C, denoted by SDC, is the possible region that exist inflection point.Fig. 3 shows the region.Here

$ \begin{array}{l} S = \left\{ {\left( {u,v} \right)\left| {uv < 0} \right.} \right\} \cup \left\{ {\left( {u,0} \right)\left| { - 1 < u < 0} \right.} \right\} \cup \\ \;\;\;\;\;\;\left\{ {\left( {0,v} \right)\left| { - 1 < v < 0} \right.} \right\}, \end{array} $

D is the open region surrounded by the curve C and the coordinate axes.

There is at least one straight line f(t0; u, v) =0 passing through every point (u0, v0) ∈SDC tangent to the curve C, here t0 is the parameter corresponding to (u0, v0).

When (u0, v0) ∈C, for $\alpha \in \left( {-\frac{3}{2}, 0} \right) $, β∈(α, 0], ${f''_{tt}}\left( {{t_0};{u_0},{v_0}} \right) = \frac{{ - 2\left( {3 + 2\alpha } \right){{\left( {\beta - \alpha } \right)}^2}}}{{2\left( {6 + 5\alpha - 3\beta } \right){t_0}\left( {1 - {t_0}} \right) + \left( {\beta - \alpha } \right)}} \ne 0. $

From

$ f\left( {t;{u_0},{v_0}} \right) = \frac{1}{2}{{f''}_{tt}}\left( {{t_0};{u_0},{v_0}} \right){\left( {t - {t_0}} \right)^2} + o{\left( {t - {t_0}} \right)^2}, $

we can see that the sign of f(t; u0, v0) does not change when it passes through t0. It means p(t0) is not an inflection point, and there is no inflection point on the curve p(t).

When (u0, v0) ∈SD, we have ft(t0; u0, v0) ≠0, otherwise (u0, v0) ∈C. From

$ f\left( {t;{u_0},{v_0}} \right) = {f'_t}\left( {{t_0};{u_0},{v_0}} \right)\left( {t - {t_0}} \right) + o\left( {t - {t_0}} \right), $

we can see that the sign of f(t; u0, v0) changes when it passes through t0. It indicates that p(t0) is an inflection point.Furthermore, there is only one straight line tangent to the curve C and passing through (u0, v0) ∈S, the corresponding curve p(t) has a single inflection point.There are two straight lines tangent to C and passing through (u0, v0) ∈D, the corresponding curve p(t) has double inflection points.

3.4 The case of convexity

We discuss the case of (u, v) ∈N=R2\(CSDL), see fig. 3. At this point, there is no cusp or loop or inflection point on the curve p(t). And, the direction of the binormal vector γ(t) does not change.

Let

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{m}}\left( t \right) = \mathit{\boldsymbol{p'}}\left( 0 \right) \wedge \left[ {\mathit{\boldsymbol{p}}\left( t \right) - \mathit{\boldsymbol{p}}\left( 0 \right)} \right],\\ \mathit{\boldsymbol{n}}\left( t \right) = \left[ {\mathit{\boldsymbol{p}}\left( t \right) - \mathit{\boldsymbol{p}}\left( 0 \right)} \right] \wedge \mathit{\boldsymbol{p'}}\left( t \right). \end{array} \right. $

According to [12], the curve p(t) is global convex, if the direction of γ(t), m(t) and n(t) do not change when they pass through t0(0 < t0 < 1). The curve is locally convex, if the direction of γ(t) does not change while that of m(t) and n(t) change when they pass through t0(0 < t0 < 1).

From eq. (5), we obtain

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{m}}\left( t \right) = \varphi \left( {t;u,v} \right)\left( {{\mathit{\boldsymbol{a}}_1} \wedge {\mathit{\boldsymbol{a}}_3}} \right),\\ \mathit{\boldsymbol{n}}\left( t \right) = \psi \left( {t;u,v} \right)\left( {{\mathit{\boldsymbol{a}}_1} \wedge {\mathit{\boldsymbol{a}}_3}} \right), \end{array} \right. $

where

$ \begin{array}{l} \varphi \left( {t;u,v} \right) = b{_2^3 }{'}\left( 0 \right)\left\{ {\left( {1 + u} \right)b_3^3\left( t \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {v\left[ {1 - 2b_2^3\left( 0 \right) - b_1^3\left( t \right)} \right]} \right\}, \end{array} $ (15)
$ \begin{array}{l} \psi \left( {t;u,v} \right) = \left[ {b_2^3\left( 0 \right) - b_0^3\left( t \right)} \right]b{_3^3 }{'}\left( t \right) + b_3^3\left( t \right)b{_0^3 }{'}\left( t \right) + \\ \;\;\;\;\;u\left\{ {\left[ {b_2^3\left( t \right) + b_3^3\left( t \right) - b_2^3\left( 0 \right)} \right]b{{_3^3}{'} }\left( t \right) - \left[ {b{{_2^3}{'} }\left( t \right) + } \right.} \right.\\ \;\;\;\;\;\left. {\left. {b{{_3^3}{'} }\left( t \right)} \right]b_3^3\left( t \right)} \right\} + v\left\{ {\left[ {b_2^3\left( t \right) + b_3^3\left( t \right) - b_2^3\left( 0 \right)} \right]b{{_0^3}{'} }\left( t \right) + } \right.\\ \;\;\;\;\;\left. {\left[ {b{{_2^3}{'} }\left( t \right) + b{{_3^3}{'} }\left( t \right)} \right]\left[ {b_2^3\left( 0 \right) - b_0^3\left( t \right)} \right]} \right\}. \end{array} $

We can see from eq.(15) that φ(t; u, v) =0 determines a family of straight lines passing through (-1, 0) on uv-plane. The slope is $k\left( t \right) =-\frac{{b_3^3\left( t \right)}}{{1-2b_2^3\left( 0 \right)-b_1^3\left( t \right)}} \in \left( {\frac{{{v^*}}}{{1 + {u^*}}}, 0} \right) $. Notice that the slope of l2 is ${k_{{l_2}}} = \frac{{{v^*}}}{{1 + {u^*}}} $. Hence, the region swept by the family of straight line φ(t; u, v) =0 in N, denoted by N1, is just the open region bounded by L1 (see eq.(10)) and l2 (see eq.(14)), see fig. 3. When (u0, v0) ∈N1, we have φt(t0; u0, v0) ≠0, otherwise (u0, v0) =(-1, 0) ∉N1 can be deduced.Then from

$ \varphi \left( {t;{u_0},{v_0}} \right) = {\varphi '_t}\left( {{t_0};{u_0},{v_0}} \right)\left( {t - {t_0}} \right) + o\left( {t - {t_0}} \right), $

we can see that the sign of φ(t; u0, v0) changes when it passes through t0. So p(t) is local convex if (u, v) ∈N1.

Solving the equations about u and v:ψ(t; u, v) =0, ψt(t; u, v) =0, the result is exactly the parameter equation of L1. Denote the region swept by the tangent line of L1 in N by N2. Then N2 is the open region bounded by L2 (see eq.(11)) and l1 (see eq.(14)), see fig. 3. When (u0, v0) ∈N2, we have ψt(t0; u0, v0) ≠0, otherwise, (u0, v0) ∉N2 can be deduced.Then from

$ \psi \left( {t;{u_0},{v_0}} \right) = {\psi '_t}\left( {{t_0};{u_0},{v_0}} \right)\left( {t - {t_0}} \right) + o\left( {t - {t_0}} \right), $

we can see that the sign of ψ(t; u0, v0) changes when it passes through t0. Hence, p(t) is local convex if (u, v) ∈N2.

While if (u, v) ∈N0=N\(N1N2), the direction of γ(t), m(t) and n(t) do not change, as a result the curve p(t) is global convex.

3.5 Result

Summarizing the discussion of section 3.1 to 3.4, we obtain the following conclusion.

Theorem 3 For αβ curve segment, let ai=PiPi-1, i=1, 2, 3, if a1 not parallel to a3 and a2=ua1+va3, then the shape characteristic of p(t) is totally determined by the distribution of the point (u, v) on uv-plane (see fig. 3), i.e.

$ \left( {u,v} \right) \in \left\{ \begin{array}{l} {N_0}\left( {{\rm{include}}\;{\rm{the}}\;{\rm{boundary}}\;{l_1} \cup {l_2} \cup \left\{ {\left. {\left( {u,0} \right)} \right|} \right.} \right.\\ \;\;\;\left. {\left. {u \ge 0\;{\rm{or}}\;u \le 1} \right\} \cup \left\{ {\left( {0,v} \right)\left| {v \ge 0\;{\rm{or}}\;v \le 1} \right.} \right\}} \right):\\ \mathit{\boldsymbol{p}}\left( t \right){\rm{is}}\;{\rm{a}}\;{\rm{global}}\;{\rm{convex}}\;{\rm{curve}}\;{\rm{and}}\;{\rm{has}}\;{\rm{no}}\;{\rm{singularity}}\\ \;\;\;\;{\rm{or}}\;{\rm{inflection}}\;{\rm{point;}}\\ {N_1} \cup {N_2}:\mathit{\boldsymbol{p}}\left( t \right)\;{\rm{is}}\;{\rm{a}}\;{\rm{local}}\;{\rm{convex}}\;{\rm{curve}}\;{\rm{and}}\\ \;\;\;{\rm{has}}\;{\rm{no}}\;{\rm{singularity}}\;{\rm{or}}\;{\rm{inflection}}\;{\rm{point;}}\\ S\left( {{\rm{include}}\;{\rm{the}}\;{\rm{boundary}}\left\{ {\left( {u,0} \right)\left| { - 1 < u < 0} \right.} \right\} \cup } \right.\\ \;\;\;\;\;\left. {\left\{ {\left( {0,v} \right)\left| { - 1 < v < 0} \right.} \right\}} \right):\mathit{\boldsymbol{p}}\left( t \right){\rm{has}}\;{\rm{one}}\\ \;\;\;\;\;{\rm{inflection}}\;{\rm{point}}\;{\rm{and}}\;{\rm{no}}\;{\rm{singularity;}}\\ D:\mathit{\boldsymbol{p}}\left( t \right){\rm{has}}\;{\rm{two}}\;{\rm{inflection}}\;{\rm{points}}\;{\rm{and}}\;{\rm{no}}\;{\rm{singularity;}}\\ C:\mathit{\boldsymbol{p}}\left( t \right){\rm{has}}\;{\rm{one}}\;{\rm{cusp}}\;{\rm{and}}\;{\rm{no}}\;{\rm{loop}}\;{\rm{or}}\;{\rm{inflection}}\;{\rm{point}};\\ L\left( {{\rm{include}}\;{\rm{the}}\;{\rm{boundary}}\;{L_1} \cup {L_2}} \right):\mathit{\boldsymbol{p}}\left( t \right){\rm{has}}\;{\rm{one}}\\ \;\;\;\;{\rm{loop}}\;{\rm{and}}\;{\rm{no}}\;{\rm{cusp}}\;{\rm{or}}\;{\rm{inflection}}\;{\rm{point}}{\rm{.}} \end{array} \right. $

Fig. 3 gives the shape diagram of the αβ curve with α=-1, β=0. It is exactly the shape diagram of the traditional cubic uniform B-spline curve.

Examples of the αβ approximation curves with the six different shape characteristics are shown in fig. 4. The settings of the parameters and the values of (u, v) are as follows:

Fig. 4 The αβ curves with different shape characteristics

(a) $\alpha =-\frac{1}{2} $, β=0, $ u = v =-\frac{2}{3}$, it is a global convex curve;

(b) $\alpha =-\frac{6}{5} $, $\beta =-\frac{1}{{10}} $, $u =-\frac{1}{5} $, $v =-\frac{1}{2} $, it is a local convex curve;

(c) $ \alpha =-\frac{6}{5}$, $\beta =-\frac{1}{{10}} $, u=1, v= -2, it has one inflection point;

(d) α=-1, β=0, $ u = v =-\frac{1}{{12}}$, it has two inflection points;

(e) α= -1, β=0, $u = v =-\frac{1}{6} $, it has one cusp;

(f) $ \alpha =-\frac{1}{2}$, β=0, $ u = v =-\frac{3}{{38}}$, it has one loop.

If a1 parallel to a3, then after a similar discussion to section 3.1 to 3.4, we obtain the following conclusion.

Theorem 4 When a1a3, the curve p(t) has no singularity.If and only if the direction of a1 is the same as that of a3, excluding the four control points collinear, the curve p(t) has one and only one inflection point.

3.6 Regulatory effect of the shape parameters

Changing α and β, the shape diagram of the αβ curve will change accordingly, see fig. 5. In fig. 5, the settings of the parameters are as follows:

Fig. 5 The influence on shape diagram by α and β

(a) $\alpha = - \frac{7}{5},\beta = 0 $; (b) $\alpha = - \frac{4}{5},\beta = 0 $;

(c) $\alpha = - \frac{1}{5},\beta = 0 $; (d) $\alpha = - \frac{6}{5},\beta = - \frac{{11}}{{10}} $;

(e) $\alpha = - \frac{6}{5},\beta = - \frac{3}{5} $; (f) $\alpha = - \frac{6}{5},\beta = - \frac{1}{{10}} $.

As can be seen from fig. 5(a) to (c), with the increase of α, the region D reduces and the region N0 expands, while the region S remains unchanged.As can be seen from fig. 5(d) to (f), with the increase of β, the region D reduces and the regions N1, N2, L expand, while the regions N0 and S remain unchanged.In addition, from fig. 5(d), we see that when βα, the regions N1, N2, L reduce to zero.From fig. 5(c), we see that when α→0 and β=0, the regions N1, N2, L and D reduce to zero.

From the above analysis, we can conclude that the region S cannot be changed by adjusting the shape parameters.That is to say, when there is only one inflection point on the αβ approximation curve, we cannot eliminate it by altering the values of α and β. However, when there is one cusp or one loop on the curve, or when the curve is local convex, it can be adjusted to a curve with two inflection points by taking βα. Further, the two inflection points can be removed by taking α→0 and β=0. Meanwhile, the curve is adjusted to global convex.

Fig. 6 shows the result of adjusting the second, the fourth, the fifth and the sixth curves in fig. 4 to global convex curves.In the figure, the black curves are the original curves in fig. 4, the red curves in the first three graphs are generated by taking $\alpha =-\frac{1}{5} $, β=0, and the red curve in the last graph is generated by taking $\alpha =-\frac{1}{{10}} $, β=0.

Fig. 6 Adjusting the αβ curves to global convex
4 The αβ surfaces

Definition 3 Given control points PijR3(i=0, 1, …, m; j=0, 1, …, n), two sets of knots u1 < u2 < … < um-1, v1 < v2 < … < vn-1, we can define (m-2) ×(n-2) αβ surface patches

$ {\mathit{\boldsymbol{p}}_{ij}}\left( {s,t} \right) = \sum\limits_{k = 0}^3 {\sum\limits_{l = 0}^3 {{\mathit{\boldsymbol{P}}_{i + k - 1,j + l - 1}}{b_k}\left( {s;{\alpha ^u},\beta _i^u} \right){b_l}\left( {s;{\alpha ^v},\beta _j^v} \right),} } $

where s, t∈[0, 1], i=1, 2, …, m-2;j=1, 2, …, n-2. All the patches constitute an αβ surface

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{q}}\left( {u,v} \right) = {\mathit{\boldsymbol{p}}_{ij}}\left( {\frac{{u - {u_i}}}{{{u_{i + 1}} - {u_i}}},\frac{{v - {v_j}}}{{{v_{j + 1}} - {v_j}}}} \right),}\\ {u \in \left[ {{u_i},{u_{i + 1}}} \right],v \in \left[ {{v_j},{v_{j + 1}}} \right],} \end{array} $

where i=1, 2, …, m-2;j=1, 2, …, n-2.

The αβ surfaces have properties similar to the αβ curves.Fig. 7 shows the αβ surfaces consisting of 7×4 patches defined by the same control net but different parameters.The settings of parameters are as follows:

Fig. 7 The αβ surfaces

(a) ${\alpha ^u} = {\alpha ^v} = - 1,\beta _i^u = \beta _j^v = 0\left( {i = 1,2, \cdots ,7;j = 1,2, \cdots ,4} \right);$

(b) ${\alpha ^u} = - \frac{7}{5},{\alpha ^v} = - \frac{{11}}{{10}},\beta _i^u = - \frac{2}{5}\left( {i = 1,2, \cdots ,7} \right),\beta _j^v = - \frac{1}{{10}}\left( {j = 1,3, \cdots ,4} \right); $

(c) ${\alpha ^u} = - \frac{6}{5},{\alpha ^v} = - \frac{{13}}{{10}},\beta _1^u = \beta _4^u = \beta _7^u = - \frac{1}{5}, $$\beta _2^u = \beta _6^u = - \frac{1}{2},\beta _3^u = \beta _5^u = - 1,\beta _1^v = \beta _3^v = 0, $$\beta _2^v = \beta _4^v = - \frac{3}{{10}}. $

5 Conclusion

There are many extensions of cubic B-spline curves, but the αβ curves enjoy some advantageous properties from design.With variation diminishing and convexity-preserving, the αβ curves have a good shape control.Shape parameters are incorporated into the αβ curves, but we do not increase the degree of the basis polynomial, and therefore do not increase the amount of calculation.The shape diagram is useful for classifying and modifying the shape of the αβ curve.Notice that if α=0, then the αβ curve segments interpolate to the two inner control points.Thus the αβ curves can be used to construct interpolation curves without solving equations.One of our future works is to discuss the shape parameter selection scheme of the interpolation curve.

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