A quartic trigonometric interpolatory spline with local free parameters

Abstract

The construction of trigonometric interpolatory splines plays a very important role in geometric modeling. This paper presents a quartic trigonometric interpolatory spline with local free parameters. The new spline not only automatically interpolates the given points and achieves C² continuity, but also owns shape adjustability when the points remain fixed. Some examples show that the shape of the new spline can easily realize local and global adjustment by changing the free parameters.

Introduction

In geometric modeling, curves are usually constructed on the basis of polynomials. However, trigonometric functions have also received very much attention within geometric modeling. Such as the trigonometric Bézier curves,^1–4 the trigonometric B-spline curves,^5–7 the trigonometric Hermite curves,^8,9 the trigonometric rational curves,^10,11 etc. The curves constructed in trigonometric function space have been applied in signal analysis, ¹² kitchen product design, ¹³ robot path planning, ¹⁴ and other engineering problems.

In recent years, the construction of trigonometric interpolatory splines, which automatically interpolate the given point, has also received attention. The two trigonometric interpolatory splines presented in Su and Tan ¹⁵ and Yan and Liang ¹⁶ provide simple and efficient ways for constructing interpolation curves. But the shape of the interpolatory spline presented in Su and Tan ¹⁵ cannot be adjusted when the points are fixed. Although the interpolatory spline presented in Yan and Liang ¹⁶ can achieve shape adjustment when the data points are fixed, it becomes G² interpolation curve. The cubic trigonometric interpolatory spline^17,18 can not only achieve shape adjustment when the points are fixed but also become C² interpolation curve. Unfortunately, the trigonometric interpolatory spline presented in Li et al. ¹⁷ and Li ¹⁸ can only achieve global adjustability when the points remain fixed, and cannot achieve local adjustability. For practical shape modeling, we often need to change the local shape of the spline. The purpose of this paper is to propose a C² quartic trigonometric interpolatory spline that can achieve local and global adjustability when the points are kept unchanged.

The main contributions of this paper are as follows:

• A quartic trigonometric spline curve that can automatically interpolate the given points is presented. The spline curve not only achieves C² continuity but also realizes local and global adjustability through the contained free parameters while keeping the points unchanged.

• The proposed spline curve is extended to the tensor product surface and the properties of the spline surface are given.

The rest of this paper is organized as follows. In Section “The basis,” the definition and some properties of the quartic trigonometric interpolatory spline basis are presented. In Section “The spline curve,” the quartic trigonometric interpolatory spline curve is defined based on the basis and some properties of the curve are given. In Section “The spline surface,” the corresponding quartic trigonometric interpolatory spline surface is presented. A short conclusion is given in Section “Conclusion.”

The basis

To define the quartic trigonometric interpolatory spline with local parameters, we first define the basis of the spline.

• Definition 1. Let $S=\sin t$ , $C=\cos t$ , $0\le t\le \pi /2$ , the basis of the quartic trigonometric interpolatory spline is expressed by

$$\{\begin{array}{c}{b}_0(\alpha ;t)=\frac{1}{6}(2(1+2\alpha )-3S-6\alpha S^2+(1-4\alpha )S^3-2(1+2\alpha )C^3+6\alpha S^4),\hfill \\ \multicolumn{1}{c}{b_1(\alpha ;t)=\frac{1}{6}(4(1-\alpha )+3C-6(1-\alpha )S^2+2(1+2\alpha )S^3-(1-4\alpha )C^3-6\alpha S^4),}\\ \multicolumn{1}{c}{b_2(\alpha ;t)=\frac{1}{6}(-2(1+2\alpha )+3S+6(1+\alpha )S^2-(1-4\alpha )S^3+2(1+2\alpha )C^3-6\alpha S^4),}\\ \multicolumn{1}{c}{b_3(\alpha ;t)=\frac{1}{6}(2(1+2\alpha )-3C-6\alpha S^2-2(1+2\alpha )S^3+(1-4\alpha )C^3+6\alpha S^4),}\end{array}$$

(1)

where α is an arbitrary real number.

• Remark 1. The spline defined in the cubic trigonometric polynomial space $\{1,\sin t,\cos t,{\sin }^{2}t,{\sin }^{3}t,{\cos }^{3}t\}$ ^17,18 has global adjustability but not local adjustability. To construct a spline that has local and global adjustability, we consider expanding the cubic trigonometric polynomial space to be quartic. Note that ${\cos }^{2}t=1-{\sin }^{2}t$ and ${\cos }^{4}t=1-2{\sin }^{2}t+{\sin }^{4}t$ always hold. To ensure that the functions in the base space are linearly independent, we leave out ${\cos }^{2}t$ and ${\cos }^{4}t$ when considering quartic trigonometric polynomial space. Then the final used quartic trigonometric polynomial space is $\{1,\sin t,\cos t,{\sin }^{2}t,{\sin }^{3}t,{\cos }^{3}t,{\sin }^{4}t\}$ .

• Theorem 1. For any value of α, the basis expressed in equation (1) has properties as follows,

• Symmetry: $b_i(\alpha ;\pi /2-t)=b_{3-i}(\alpha ;t)$ , $i=0,1,2,3$ .

• Partition of unity: $\sum _{i=0}^3{b}_i(\alpha ;t)\equiv 1$ .

• Terminal properties:

$$\{\begin{array}{c}\begin{array}{cccc}{b}_0(\alpha ;0)=0,\hfill & b_1(\alpha ;0)=1,\hfill & b_2(\alpha ;0)=0,\hfill & b_3(\alpha ;0)=0,\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cccc}{b}_0(\alpha ;\pi /2)=0,\hfill & b_1(\alpha ;\pi /2)=0,\hfill & b_2(\alpha ;\pi /2)=1,\hfill & b_3(\alpha ;\pi /2)=0.\hfill \end{array}}\end{array}$$

(2)

$$\{\begin{array}{c}\begin{array}{cccc}{b\prime }_0(\alpha ;0)=-1/2,\hfill & {b\prime }_1(\alpha ;0)=0,\hfill & {b\prime }_2(\alpha ;0)=1/2,\hfill & {b\prime }_3(\alpha ;0)=0,\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cccc}{b\prime }_0(\alpha ;\pi /2)=0,\hfill & {b\prime }_1(\alpha ;\pi /2)=-1/2,\hfill & {b\prime }_2(\alpha ;\pi /2)=0,\hfill & {b\prime }_3(\alpha ;\pi /2)=1/2.\hfill \end{array}}\end{array}$$

(3)

$$\{\begin{array}{c}\begin{array}{cccc}{b″}_0(\alpha ;0)=1,\hfill & {b″}_1(\alpha ;0)=-2,\hfill & {b″}_2(\alpha ;0)=1,\hfill & {b″}_3(\alpha ;0)=0,\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cccc}{b″}_0(\alpha ;\pi /2)=0,\hfill & {b″}_1(\alpha ;\pi /2)=1,\hfill & {b″}_2(\alpha ;\pi /2)=-2,\hfill & {b″}_3(\alpha ;\pi /2)=1.\hfill \end{array}}\end{array}$$

(4)

Proof. By simple calculations, (b) and (c) are not hard to be deduced. We just verify (a). Since $C^2=1-S^2$ and $C^4=(1-S^2{)}^2=1-2S^2+S^4$ , for any value of α we have

$$\begin{array}{c}{b}_0(\alpha ;\pi /2-t)=\frac{1}{6}(2(1+2\alpha )-3C-6\alpha (1-S^2)+(1-4\alpha )C^3-2(1+2\alpha )S^3+6\alpha (1-2S^2+S^4))\hfill \\ =\frac{1}{6}(2(1+2\alpha )-3C-6\alpha S^2-2(1+2\alpha )S^3+(1-4\alpha )C^3+6\alpha S^4)\hfill \\ =b_3(\alpha ;t).\hfill \end{array}$$

For the same reason, we can get $b_1(\alpha ;\pi /2-t)=b_2(\alpha ;t)$ . Thus, the symmetry of the basis has been proved.

From equation (1), the basis with different shapes would be obtained if the free parameter α is taken as different values. Figure 1 shows the graphs of the basis with multiple values of α.

Figure 1.

The graphs of the basis with multiple values of α: (a) $b_0(\alpha ;t)$ , (b) $b_1(\alpha ;t)$ , (c) $b_2(\alpha ;t)$ , and (d) $b_3(\alpha ;t)$ .

The spline curve

Based on the basis expressed in equation (1), we can define the corresponding quadric trigonometric interpolatory spline curve.

• Definition 2. Given a series of points $p_i\in R^k$ ( $k=2,3$ ; $i=0,1,...,n$ , $n\ge 3$ ), for $0\le t\le \pi /2$ , the curve

$$Q_i({\alpha }_i;t)=\sum _{j=0}^3{b}_j({\alpha }_i;t)p_{i+j},i=0,1,...,n-3,$$

(5)

is called quartic trigonometric interpolatory spline curve (QTI-spline curve for short), where $b_j({\alpha }_i;t)$ $(j=0,1,2,3)$ is the basis expressed according to equation (1).

• Remark 2. From Definition 2, we can see that the QTI-spline curve is composed of $n-2$ segments and each segment $Q_i({\alpha }_i;t)$ $(i=0,1,...,n-3)$ contains a free parameter ${\alpha }_i$ .

• Theorem 2. The QTI-spline curve has properties as follows,

• Interpolation. For any ${\alpha }_i$ $(i=0,1,...,n-3)$ , the QTI-spline curve interpolates the given points $p_i$ $(i=1,2,...,n-1)$ .

• Continuity. For any ${\alpha }_i$ $(i=0,1,...,n-3)$ , the QTI-spline curve satisfies C² continuity. Particularly, the QTI-spline curve reaches C³ continuity when ${\alpha }_i=-1/2$ $(i=0,1,...,n-3)$ .

• Shape adjustability. When the given points $p_i$ $(i=0,1,...,n)$ are kept unchanged, the shape of the QTI-spline curve can be adjusted locally and globally.

Proof. (a) For any ${\alpha }_i$ $(i=0,1,...,n-3)$ , by computing from equations (2) and (5) we have

$$Q_i({\alpha }_i;0)=p_{i+1},Q_i({\alpha }_i;1)=p_{i+2}.$$

(6)

Equation (6) shows that the QTI-spline curve interpolates the points $p_i$ $(i=1,2,...,n-1)$ .

(b) For any ${\alpha }_i$ $(i=0,1,...,n-3)$ , by computing from equations (3) to (5) we have

$$\frac{\mathrm{d}{Q}_i({\alpha }_i;0)}{\mathrm{d}t}=\frac{1}{2}(p_{i+2}-p_i),\frac{\mathrm{d}{Q}_i({\alpha }_i;\pi /2)}{\mathrm{d}t}=\frac{1}{2}(p_{i+3}-p_{i+1}),$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\mathrm{d}}^2{Q}_i({\alpha }_i;0)}{\mathrm{d}{t}^2}=p_{i+2}-2p_{i+1}+p_i,\frac{{\mathrm{d}}^2{Q}_i({\alpha }_i;\pi /2)}{\mathrm{d}{t}^2}}\\ =p_{i+3}-2p_{i+2}+p_{i+1}.\hfill \end{array}$$

(8)

From equations (6) to (8), we have

$$\frac{{\mathrm{d}}^k{Q}_i({\alpha }_i;\pi /2)}{\mathrm{d}{t}^k}=\frac{{\mathrm{d}}^k{Q}_{i+1}({\alpha }_{i+1};0)}{\mathrm{d}{t}^k},k=0,1,2.$$

(9)

Equation (9) shows that the QTI-spline curve satisfies C² continuity. When ${\alpha }_i=-1/2$ $(i=0,1,...,n-3)$ , by computing from equations (1) and (5) we have

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\mathrm{d}}^3{Q}_i(1/2;0)}{\mathrm{d}{t}^3}=\frac{7}{2}(p_i-p_{i+2}),\frac{{\mathrm{d}}^2{Q}_i(1/2;\pi /2)}{\mathrm{d}{t}^2}}\\ =\frac{7}{2}(p_{i+1}-p_{i+3}).\hfill \end{array}$$

(10)

From equation (10), we obtain

$$\frac{{\mathrm{d}}^3{Q}_i(1/2;\pi /2)}{\mathrm{d}{t}^3}=\frac{{\mathrm{d}}^3{Q}_{i+1}(1/2;0)}{\mathrm{d}{t}^3}.$$

(11)

Equations (9) and (11) show that the QTI-spline curve reaches C³ continuity when ${\alpha }_i=-1/2$ $(i=0,1,...,n-3)$ .

(c) Since each segment of the QTI-spline curve contains a free parameter, changing the value of this free parameter will only affect the shape of this segment, but not the shape of other segments. This means we can realize the local adjustment of the QTI-spline curve through the free parameters contained in each segment when the given points are fixed. If we set the free parameters contained in all segments as a unified parameter, we can realize the global adjustment of the QTI-spline curve by modifying this unified parameter when the given points are fixed.

• Example 1. Given the points as follows,

$$\begin{array}{c}{p}_0=p_1=(0,1),p_2=(2,0),p_3=(4,1),p_4=(7,0),\hfill \\ p_5=p_6=(8,1).\hfill \end{array}$$

Figure 2 shows the local adjustment of the planar QTI-spline curve by altering the values of the free parameters contained in the curve, where the free parameters ${\alpha }_i=1$ $(i=0,1,3)$ but ${\alpha }_2$ is taken as different values.

Figure 2.

The local adjustment of the planar QTI-spline curve.

The global adjustment of the planar QTI-spline curve by altering the values of the free parameters contained in the curve is shown in Figure 3, where the free parameters are unified to ${\alpha }_i=\alpha$ $(i=0,1,2,3)$ and $\alpha$ is taken as different values.

• Example 2. Given the points as follows,

$$\begin{array}{c}{p}_0=(0,1,0),p_1=(2,0,1),p_2=(4,1,0),p_3=(5,0,1),\hfill \\ p_4=(8,1,0),p_5=(10,1,0).\hfill \end{array}$$

Figure 3.

The global adjustment of the planar QTI-spline curve.

Figure 4 shows the local adjustment of the planar QTI-spline curve by altering the values of the free parameters contained in the curve, where the free parameters ${\alpha }_i=0$ $(i=0,2)$ but ${\alpha }_1$ is taken as different values.

Figure 4.

The local adjustment of the spatial QTI-spline curve.

The global adjustment of the spatial QTI-spline curve through the free parameters contained in the curve is shown in Figure 5, where the free parameters are unified to ${\alpha }_i=\alpha$ $(i=0,1,2)$ and $\alpha$ is taken as different values.

Figure 5.

The global adjustment of the spatial QTI-spline curve.

The spline surface

By extending the QTI-spline curve to a tensor product form, we can define the corresponding spline surface.

• Definition 3. Given a series of points $p_{i,j}\in R^3$ ( $i=0,1,...,m$ , $j=0,1,...,n$ , $m,n\ge 3$ ), for $0\le u,v\le \pi /2$ , the surface

$$\begin{array}{c}\multicolumn{1}{c}{R_{i,j}(u,v)=\sum _{k=0}^3\sum _{l=0}^3{b}_k({\alpha }_i;u)b_l({\beta }_j;v)p_{i+k,j+l},}\\ \multicolumn{1}{c}{i=0,1,...,m-3,j=0,1,...,n-3,}\end{array}$$

(12)

is called quartic trigonometric interpolatory spline surface (QTI-spline surface for short), where $b_k({\alpha }_i;u)$ and $b_k({\beta }_j;v)$ $(k=0,1,2,3)$ are the basis expressed according to equation (1).

• Remark 3. From Definition 3, we can see that the QTI-spline surface is composed of $(m-3)\times (n-3)$ patches and each patch $R_{i,j}(u,v)$ ( $i=0,1,...,m-3$ , $j=0,1,...,n-3$ ) contains two free parameters ${\alpha }_i$ and ${\beta }_j$ .

Referring to Theorem 2, we can easily obtain the properties of the QTI-spline surface. We give the properties of the QTI-spline surface without proof.

• Theorem 3. The QTI-spline surface has properties as follows,

• (a) Interpolation. For any ${\alpha }_i$ and ${\beta }_j$ ( $i=0,1,...,m-3$ , $j=0,1,...,n-3$ ), the QTI-spline surface interpolates the given points $p_{i,j}$ ( $i=1,2,...,m-1$ , $j=1,2,...,n-1$ ).

• (b) Continuity. For any ${\alpha }_i$ and ${\beta }_j$ ( $i=0,1,...,m-3$ , $j=0,1,...,n-3$ ), the QTI-spline surface satisfies C² continuity. Particularly, the QTI-spline surface reaches C³ continuity when ${\alpha }_i={\beta }_j=-1/2$ ( $i=0,1,...,m-3$ , $j=0,1,...,n-3$ ).

• (c) Shape adjustability. When the given points $p_{i,j}$ ( $i=0,1,...,m$ , $j=0,1,...,n$ ) are kept unchanged, the shape of the QTI-spline surface can be adjusted locally and globally.

• Example 3. Given the points as follows,

$$\begin{array}{c}{p}_{0,0}=(-2,-6,1),p_{0,1}=(-1,-6,1),p_{0,2}=(0,-6,1),\hfill \\ p_{0,3}=(1,-6,1),\hfill \end{array}$$

$$\begin{array}{c}{p}_{1,0}=(-2,-4,0),p_{1,1}=(-1,-4,0),p_{1,2}=(0,-4,0),\hfill \\ p_{1,3}=(1,-4,0),\hfill \end{array}$$

$$\begin{array}{c}{p}_{2,0}=(-2,-2,1),p_{2,1}=(-1,-2,1),p_{2,2}=(0,-2,1),\hfill \\ p_{2,3}=(1,-2,1),\hfill \end{array}$$

$$\begin{array}{c}{p}_{3,0}=(-2,0,1),p_{3,1}=(-1,0,1),p_{3,2}=(0,0,1),\hfill \\ p_{3,3}=(1,0,1),\hfill \end{array}$$

$$\begin{array}{c}{p}_{4,0}=(-2,2,0),p_{4,1}=(-1,2,0),p_{4,2}=(0,2,0),\hfill \\ p_{4,3}=(1,2,0),\hfill \end{array}$$

$$\begin{array}{c}{p}_{5,0}=(-2,4,1),p_{5,1}=(-1,4,1),p_{5,2}=(0,4,1),\hfill \\ p_{5,3}=(1,4,1).\hfill \end{array}$$

Figure 6 shows the local adjustment of the QTI-spline surface by altering the values of the free parameters contained in the surface, where the surface is composed of three patches, and the free parameters $({\alpha }_i,{\beta }_j)=(1,2)$ $(i=0,2,j=0)$ are remained unchanged but $({\alpha }_1,{\beta }_0)$ is taken as different values.

Figure 6.

The local adjustment of the QTI-spline surface: (a) $({\alpha }_1,{\beta }_0)=(-2,-1)$ , (b) $({\alpha }_1,{\beta }_0)=(-3,3)$ , (c) $({\alpha }_1,{\beta }_0)=(2,-1)$ , and (d) $({\alpha }_1,{\beta }_0)=(0,0)$ .

Figure 7 shows the global adjustment of the QTI-spline surface through the free parameters contained in the surface, where the free parameters are unified to $({\alpha }_i,{\beta }_j)=(\alpha ,\beta )$ $(i=0,1,2,j=0)$ , and $(\alpha ,\beta )$ is taken as different values.

Figure 7.

The global adjustment of the QTI-spline surface: (a) $(\alpha ,\beta )=(-3,-2)$ , (b) $(\alpha ,\beta )=(-6,4)$ , (c) $(\alpha ,\beta )=(3,-2)$ , and (d) $(\alpha ,\beta )=(5,3)$ .

Conclusion

We present a novel quartic trigonometric interpolatory spline with local free parameters in this paper. Different from some existing similar trigonometric interpolatory splines, the proposed C² spline is not only globally adjustable but also locally adjustable. This feature is very conducive to geometric modeling.

For practical applications of the proposed spline in geometric modeling, it is clear that some special algorithms need to be established. Some interesting results in this area will be presented in a follow up study.