Quasi Cubic Trigonometric Curve and Surface

Firstly, a new set of Quasi-Cubic Trigonometric Bernstein basis with two tension shape parameters is constructed, and we prove that it is an optimal normalized totally basis in the framework of Quasi Extended Chebyshev space. And the Quasi-Cubic Trigonometric Bézier curve is generated by the basis function and the cutting algorithm of the curve are given, the shape features (cusp, inflection point, loop and convexity) of the Quasi-Cubic Trigonometric Bézier curve are analyzed by using envelope theory and topological mapping; Next we construct the non-uniform Quasi-Cubic Trigonometric B-spline basis by assuming the linear combination of the optimal normalized totally positive basis have partition of unity and continuity, and its expression is obtained. And the non-uniform B-spline basis is proved to have totally positive and high-order continuity. Finally, the non-uniform Quasi Cubic Trigonometric B-spline curve and surface are defined, the shape features of the non-uniform Quasi-Cubic Trigonometric B-spline curve are discussed, and the curve and surface are proved to be continuous.


Introduction
The construction of basis functions has always been an important topic in computer-aided geometric design (CAGD)and computer graphics(CG). A class of practical basis functions often have an important impact on the development of the geometric industry. Traditional Bernstein and B-spline basis, especially cubic basis functions, are widely used in geometric industrial designs, but the resulting curve positions are relatively fixed to the control polygon. Although the NURBS method has a weighting factor that can adjust the shape of the curve, the effect of the weighting factor on the curve is often unclear to the researchers [1][2][3].
In order to solve such problems, a large number of new basis functions are constructed. The simplest method is to add shape parameters to the classical Bernstein and B-spline basis [4][5][6][7][8]. But in order to be able to effectively modify the shape of the curve, a lot of work is around the tension parameters [9][10][11][12][13][14]. Based on the 4 C splines, the 3 2 FC C  spline curve with tension parameters is constructed [15]. On this basis, a group of 3 2 G C  spline curve with three shape parameters is constructed [16]. In [17], Han constructed a class of spline curves with two exponential shape parameters which possesses continuity. Given the shape preserving property of the curve, Costantini [18] constructed a set of variable degree polynomial basis in the space } , , which is a Quasi Extended Chebyshev(QEC) space [19,20]. From that, the space } , ) 1 ( , , , is also proved to be a QEC space [21]. Lately, in the framework of Canonical Complete Chebyshev space, it is proved that the variable degree splines basis possesses totally positivity property [22]. In [23], Zhu constructed a kind of quasi Bernstein basis with two shape parameters, which is proved an optimal normalized totally positive basis, and the related B-spline curve possessing continuous. In [24], Wang constructed a group of DTB-like basis with two denominator shape parameters, and the associated B-spline basis possesses 1 2 − n C continuity when the shape parameters are globally parameter. In [36], the changeable spline basis is given.
In this paper, the theoretical knowledge of QEC-space is applied to prove that  The other work arrangements of this paper are as follows: A class of B basis is given in Section 3. Section 4 presents the definition, corner cutting algorithm and shape features of QCT-Bézier curve. In the section 5, a group of QCT-B spline basis is proposed and its properties are analyzed. Section 6 gives the definition and local adjustable properties of QCT-B spline curve. Section 7 gives the definition and high order continuity of QCT-B spline surfaces. Finally, Section 8 gives the conclusion.
Use I to represent an arbitrarily given ] , [ b a , which has the following definition:

Construction of B basis
It can be known from the theorem 3.1 of [24] that it is only necessary to prove that the related differential space  Therefore,

DT
is a 3-dimension space.
Next, we will prove that Therefore, according to Wronskian, we have , we define 3 weight functions as follows , are 3 any positive real numbers. And We consider the following ECC space defined by We can verify that ) ( ), ( . From the precious proof, we have obtained that . Therefore, we can get that ) (t g is monotonically decreasing or monotonically increasing function on We also call B basis as QCT-Bernstein basis with two parameters  and  .

Proof: For any
From the properties of blossom [16][17][18][19][20][21], we have , we can easily deduce the expressions of (3). Next, we verify that , we consider the following linear combination: And then we differentiate both side of (4), we have When (4) and (5) , we have is called a QCT-Bézier curves with 2 shape parameters  and  .
Since QCT-Bernstein basis has the properties of totally positive, nonnegatively and partition of unity, the related QCT-Bézier curves devoted in (6) possesses variation diminishing, affine invariance and convex hull, which means that QCT-Bézier curves are suitable for geometric design. In addition, for any , we have the following end-point property: From the properties discussed above, we know that the QCT-Bézier curves keep all the properties of traditional Bézier curves. Since QCT-Bézier curves have 2 parameters  and  , it has the property of flexible shape adjustability.

Shape adjustment of QCT-Bézier curves
Next, we rewrite (6) as the following form: Obviously, for any settled decreases with the increase of  , which indicates that the QCT-Bézier curve has the same direction of the edge vector 1 0 P P − as  increases. When  decreases, the situation is just the opposite. For edge vector 2 3 P P − ,  has the same effect. When  or  increases, QCT-Bézier curves will approach 2 P and 3 P , respectively. When   = increases or decreases, QCT-Bézier curve has the opposite or same direction of the edge vector 1 2 P P − , which indicates that parameters  and  have the tension effect. Fig.2 shows the QCT-Bézier curves with different shape parameters.

The shape analysis of QCT-Bézier curves
In this section, we will use envelope theory and topological mapping to describe the shape features of the proposed QCT-Bézier curve given in (6). The relevant cusps, inflection points, loops, convexity as well as envelope and topological mapping theories can be referred to [33][34][35] Thus, we have are not coplanar, we can know that the vector , and it has not cusps.
Next, we assume that ) (t Q possesses loops, when According precious discussion ,we have known that . Then, we let has not loops.
is mixed product of vector edge , by directly computing, we have 0 For , and it has same positive and negative property as . Thus, ) (t Q has not inflection points, and it has the same rotation direction as the control points.

The shape analysis of the planar QCT-Bézier curve
. Firstly, we Substituting it into (9), we have

1) Cusps
The necessary condition that the planar QCT-Bézier curve Because 1 a and 3 a are linear independent, from (12) and (3) , we have We analyze the shape of C , from (13), we have This shows that C has two asymptotes .This indicates that the C is a monotonically decreasing and strictly convex curve. For any . In fact, similar to the discussion of (12) and (13), we have For any , we can easily verify that (13) and (15) cannot be established at the same time. It indicates will changes. We can easily conclude that ) ( 0 t Q is a cusp. Therefore, the planar QCT-Bézier curve defined by (11) possess cusps are equivalent to C v u  ) , ( .

2) Inflection points
From the previous discussion , we have is an inflection point if and only if ) , ; ( v u t f change sign at 0 t . In the − uv plane, the possible region of inflection points must be covered by the family of straight lines 0 ) , . According to [33], the curve C is just the envelope of the family of straight lines. From the previous analysis, it can be known that the curve C is a strictly convex continuous curve. Thus, the area swept by the tangent line of the curve C is , that is, the area where the inflection point may occur. In Fig. 4, ' S ' represents the area where the QCT-Bézier curve has one single inflection point; '  , passing through an any point does not changes sign at 0 t , the planar QCT-Bézier curve has not inflection points. When We can easily get 0 ) , , ( from the definition of envelope, we can know that will change sign at 0 t , the planar QCT-Bézier curve ) (t Q has a inflection point at 0 t . In addition, when , the curve C only has a tangent line which passes ) only has one inflection point; When has two inflection points.

3) Loops
The planar QCT-Bézier curve ) (t Q has loops if and only if there exists . This is equivalent to the parameter satisfying the following equations: . Easy to verify (17) defines a topological mapping From the mathematical analysis, it can be deduced that the curve

4) Convexity
, the planar QCT-Bézier curve ) (t Q has not cusps, inflection points and loops. Next, we consider the following vector According to (11) and (20), by direct computing, we have changes the direction at 0 t . We can easily get Fig.4), the curve ) (t Q is locally convex [33]. In fact, Therefore, when

5) Adjusting effect of shape parameters
Through the above analysis of the shape of the QCT-Bézier curve, the following conclusions can be drawn: ① As shown in Fig.5, the change of the shape parameter  and  does not affect the single inflection point region S and the global convex region 0 N , so when ) (t Q is global convex, it cannot be eliminated by adjusting parameters; and when it is global convex, the shape parameters are modified anyway, the curve is still global convex. ② As the shape parameters  and  increase, the curve 2 L is stretched toward the point (0,-1), double inflection points region D shrinks, loops region L expands correspondingly, and locally convex regions 1 N and 2 N expand, as shown in Fig.5.
, i.e. when the first and last two edges of the polygon are intersected (except the first and last points coincide), the curve ) (t Q may has cusps and loops or inflection points. The curve ) (t Q cannot be made into a locally convex curve simply by modifying the shape parameter, as shown in Fig. 6.
,i.e. when the control polygon is locally convex, the cusps, inflection points and loops of the curve ) (t Q can be eliminated by modifying the shape parameter, and the curve can be adjusted to a locally convex curve, as shown in Fig.7.

The construction of QCT-B spline basis
Given knots spline basis is constructed as follows: is the QCT-Bernstein basis given in equation (3).
In order to determine the coefficient , we call (9) is QCT-B spline basis with 2 shape parameters.
For equidistant knots, the QCT-B spline basis is called a uniform QCT-B spline basis, and the corresponding knot vector is called an equidistant knot vector. For non-uniform knots, the QCT-B spline basis is called a non-uniform QCT-B spline basis, and the corresponding knot vector is called a non-equidistant knot vector. Fig.8 shows the image of the uniform QCT-B-spline basis under different shape parameters. Fig.8 Uniform B-spline basis Direct calculations yield the following lemma, which will be very useful for subsequent discussions. Lemma 1 For any Z i  , the following equation is true: , by simplifying the basis function (22) from two parameters to one parameter, the corresponding coefficient can be simplified to as follows: . Given that the system )) ;   (   1  2  1   1  1  1  2  2  1   2  1  1  1  1   1  1  1  1  1   1  1  1  1  1  2  1  1   1  1  1   3  ,  3  ,   0 , 0 , Proof First, we use verify that the ) 1 2 ( − n -order derivative of (3) has the following form: Thus, The form (23) is meet when 1 = n .
We assume that the form (23) is also meet when k n = . Therefore, the ) We can easily find that the form (23) as a non-uniform QCT-B spline curve with two shape parameters.
Obviously, for , the QCT-B spline curve segment curve can be expressed as following form Available from Theorem 5 and 6, for Thus, we have , ) ( According to the Theorem 9 and the coefficients , so we can easily obtain Fig.9 The QCT-B spline curve

Local adjustable properties
The non-uniform QCT-B spline curve presented in the paper possesses two shape parameters

Shape analysis of QCT-B spline curves
In this section, we will use the envelope theory and topological mapping theory to describe the shape features of the non-uniform QCT-B spline curve segments given in (26).

Shape analysis of spatial QCT-B spline curves
are not coplanar, the QCT-Bézier curve segments are spatial curve, it does not have cusps, loops and inflection points, and it has the same rotation direction as the control points.
, the QCT-Bézier curve segments given in (26) can be rewrite as are not coplanar, the knot vector ) , are linear independent, and . Therefore, we can easily get 0 ) ( . Obviously, this can lead to contradictions. Thus, we can get , and we also can easily get the curve segments ) (u Q i have not cusps and inflection points. Next, we consider the inflection points. Let )) ( ), 2   1  1  1  1   2  2  2  2   3  3  3  3   1  2  3   1  1  1  1   2  2  2  2   3  3  3  3   1  2  3   3  3  3  3   3  3  3  3   3  3 , and it has same positive and negative property as has not inflection points, and it has the same rotation direction as the control points. , let  Where, the description of each distribution area is as follows: D is an open region surrounded by coordinate the axis V U, and curve C . L is an region surrounded the curve . The parametric equations for the relevant curves are as follows: , we can easily get , then we combine (29), we have Below we discuss the cusps, inflection points, loops and convexity.

1) Cusps
The necessary condition that the planar QCT-B spline curve are linear independent, according to (36) , we can get the curve C : We analyze the shape of the curve C given in (30) and (37)  . In fact, similar to the discussion of (36) and (37), change its sign when the curve passes the point 0 u . Where, Since 0

3) Loops
The non-uniform QCT-B spline curve has loops means that ) ( ) ( . This is equivalent to satisfying the following equations: . The equations given in (40) define a topological mapping is a simply connected region in − UV plane. The three boundary lines of L correspond to the three boundary lines  (33) and (34), respectively.

4) Convexity
The following is the case for ) . Obviously, the curve ) (u Q i has not cusps, inflection points and loops. Next, we consider )] determines a family of straight lines passing ) 0 , 1 (− on the UV -plane. , therefore, the region swept by the line family 0 ) , in N happens to be the part enclosed by the curve 1 L and the straight line segment 1 l , which we record as 1 N (excluding the boundary lines 1 L and 1 l , as shown in Fig. 10). If ). Thus, from the expansion ) ( ) )( , ; ( ) , ; ( , we can know that ) , ; (   L gradually shrinks, and the global convex region Z gradually expands, see [11].

Preprints
In fact, in addition to variation diminishing, the properties of QCT-B spline curves can be extended to QCT-B spline surface. Due to space limitation, we will not discuss it in detail here, but the higher-order continuity of uniform QCT-B spline surface is given below.

Theorem 15
Given two uniform knot vector U and V , when shape parameters satisfy 

Conclusion
When the traditional literature improves the Bézier method and the B-spline method, it only focuses on whether the curve flexibility can be increased. Therefore, the constructed curves and surfaces retain only some basic properties of the Bézier method and the B-spline method, such as convex hull and affine invariance, symmetry, etc., such as totally positive property, variation diminishing property and shape preservation are overlooked. In view of the problems of traditional improved methods, this paper constructs a set of optimal norm-positive basis from the property of shape preservation, and designs curve and surfaces with high-order continuity. A lot of discussion and analysis show that the curve and surface constructed in this paper not only retains the good properties of the traditional Bézier method and the B-spline method, but also has shape preservation , shape adjustability and high-order continuity, which is suitable for curve and surface design. In addition, the shape of the curve, such as sharp points, inflection points, loops, convexity, etc., is analyzed in detail, which will further facilitate the design of better geometric shapes. Although the proposed method has many advantages, there are still many problems that have not been solved, such as the reverse problem of parameters, how to extend the curve to the triangular domain, etc., which will be the future work.