2. College of Science, China Three Gorges University, Yichang 443002, China
In geometric modeling, curves are usually constructed on the basis of polynomials. Recently, trigonometric polynomials have received much attention within geometric modeling. Some examples are the trigonometric polynomial Bézier curves [1][4], the trigonometric polynomial Bspline curves [5][8], and the trigonometric Hermite spline curves [9], [10]. Those trigonometric polynomial curves not only inherit similarities of the corresponding classical polynomial curves, but also have better performance ability in different aspects.
It is well known that the spline interpolation has a wide range of applications in geometric modeling. However, when the traditional cubic splines are used to construct
The rest of this paper is organized as follows. In Section Ⅱ, the cubic trigonometric automatic interpolation spline curves with two parameters are constructed, and properties of interpolation spline curves are discussed. In Section Ⅲ, a method for determining the optimal automatic interpolation spline curves is presented, and some examples are given. A short conclusion is given in Section Ⅳ.
Ⅱ. THE CUBIC TRIGONOMETRIC AUTOMATIC INTERPOLATION SPLINE CURVES A. The Basis FunctionsFirstly, the cubic trigonometric spline basis functions with two parameters are defined as follows.
Definition 1: For
$ \begin{align} \label{eq1} \begin{cases} f_{0} (t)=\dfrac{1}{24}((14\alpha 2\beta +6)+(9\alpha +3\beta 9)S+24\alpha S^{2}\\[1mm] \qquad\quad~ +(19\alpha \beta +3)S^{3}+(14\alpha +2\beta 6)C^{3} ) \\[3mm] f_{1} (t)=\dfrac{1}{24}((2\alpha 10\beta +6)+(9\alpha 3\beta +9)C+24\beta S^{2}\\[1mm] \qquad\quad~+(2\alpha 14\beta 6)S^{3}+(7\alpha +13\beta +9)C^{3} ) \\[3mm] f_{2} (t)=\dfrac{1}{24}((2\alpha +14\beta +6)+(9\alpha 3\beta +9)S24\beta S^{2}\\[1mm] \qquad\quad~+(7\alpha +13\beta +9)S^{3}+(2\alpha 14\beta 6)C^{3} ) \\[3mm] f_{3} (t)=\dfrac{1}{24}((10\alpha 2\beta +6)+(9\alpha +3\beta 9)C24\alpha S^{2}\\[1mm] \qquad\quad~+(14\alpha +2\beta 6)S^{3}+(19\alpha \beta +3)C^{3} ) \end{cases} \end{align} $  (1) 
where
By simple deduction, the cubic trigonometric spline basis functions defined as (1) have the following properties at the endpoints,
$ \begin{cases} {f_{0} (0)=0, } ~~ {f_{1} (0)=1, } ~~ {f_{2} (0)=0, } ~~ {f_{3} (0)=0} \\ {f_{0} (1)=0, } ~~ {f_{1} (1)=0, } ~~ {f_{2} (1)=1, } ~~ {f_{3} (1)=0} \\ \end{cases} $  (2) 
$ \begin{cases} {{f}'_{0} (0)=\dfrac{\pi} {16}\left({3\alpha \beta +3} \right), } ~~ {{f}'_{1} (0)=0} \\[7pt] {{f}'_{2} (0)=\dfrac{\pi} {16}\left({3\alpha \beta +3} \right), } ~~ {{f}'_{3} (0)=0} \\[7pt] {{f}'_{0} (1)=0, } ~~ {{f}'_{1} (1)=\dfrac{\pi} {16}\left({3\alpha \beta +3} \right)} \\[7pt] {{f}'_{2} (1)=0, } ~~ {{f}'_{3} (1)=\dfrac{\pi} {16}\left({3\alpha \beta +3} \right)} \\ \end{cases} $  (3) 
$ \begin{cases} {{f}''_{0} (0)=\dfrac{\pi^{2}}{16}\left({\alpha \beta +3} \right), } ~~ {{f}''_{1} (0)=\dfrac{\pi^{2}}{8}\left({\alpha \beta +3} \right)} \\[7pt] {{f}''_{2} (0)=\dfrac{\pi^{2}}{16}\left({\alpha \beta +3} \right), } ~~ {{f}''_{3} (0)=0} \\[7pt] {{f}''_{0} (1)=0, } ~~ {{f}''_{1} (0)=\dfrac{\pi^{2}}{16}\left( {\alpha \beta +3} \right)} \\[7pt] {{f}''_{2} (0)=\dfrac{\pi ^{2}}{8}\left({\alpha \beta +3} \right), } ~~ {{f}''_{3} (0)=\dfrac{\pi^{2}}{16}\left({\alpha \beta +3} \right)}. \\ \end{cases} $  (4) 
From (1), the cubic trigonometric spline basis functions have two free parameters
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Fig. 1 Curves of the proposed basis functions for different 
Depending on the cubic trigonometric spline basis functions, the cubic trigonometric automatic interpolation spline curves with two parameters can be defined as follows.
Definition 2: Given a series of data points
$ \begin{align} \label{eq5} S_{i} (x)=&\ f_{0} \left({\frac{xx_{i}} {h_{i}}} \right)y_{i1} +f_{1} \left({\frac{xx_{i}} {h_{i}}} \right)y_{i} \nonumber\\[2mm] & +f_{2} \left({\frac{xx_{i} }{h_{i}}} \right)y_{i+1} +f_{3} \left({\frac{xx_{i}} {h_{i}}} \right)y_{i+2} \end{align} $  (5) 
where
Theorem 1: The CTAIspline curves defined as (5) have the following properties,
a) Automatic interpolation property: For given data points
$ \begin{align*} \begin{cases} S_{i} (x_{i})=y_{i} \\[1mm] S_{i} (x_{i+1})=y_{i+1} \\ \end{cases}, \quad i=1, 2, \ldots, n2. \end{align*} $ 
b)
$ \begin{align*} S_{i}^{(k)} (x_{i+1})=S_{i+1}^{(k)} (x_{i+1}), \quad k=0, 1, 2;~ i=1, 2, \ldots, n3. \end{align*} $ 
Proof: a) By (2) and (5), we have
$ \begin{align} \label{eq6} \begin{cases} S_{i} (x_{i})=f_{0} (0)y_{i1} +f_{1} (0)y_{i} +f_{2} (0)y_{i+1} +f_{3} (0)y_{i+2} \\ \qquad~~=y_{i} \\[2mm] S_{i} (x_{i+1})=f_{0} (1)y_{i1} +f_{1} (1)y_{i} +f_{2} (1)y_{i+1} +f_{3} (1)y_{i+2} \\ \qquad\quad~~=y_{i+1} \\ \end{cases} \end{align} $  (6) 
Equation (6) shows that the CTAIspline curves
b) When
$ \begin{align} \label{eq7} \begin{cases} {S}'_{i} (x_{i})=\dfrac{1}{h}\left({{f}'_{0} (0)y_{i1} {+}{f}'_{1} (0)y_{i} +{f}'_{2} (0)y_{i{+}1} {+}{f}'_{3} (0)y_{i{+}2}} \right)\\[3mm] \qquad~~=\dfrac{\pi }{16h}(3\alpha \beta +3)~(y_{i+1} y_{i1}) \\[5mm] {S}'_{i} (x_{i{+}1})=\dfrac{1}{h}\left({{f}'_{0} (1)y_{i1} {+}{f}'_{1} (1)y_{i} {+}{f}'_{2} (1)y_{i{+}1} {+}{f}'_{3} (1)y_{i{+}2}} \right)\\[3mm] \qquad\quad~~=\dfrac{\pi }{16h}(3\alpha \beta +3)~(y_{i+2} y_{i}). \end{cases} \end{align} $  (7) 
By (4) and (5), we have
$ \begin{align} \label{eq8} ~~~~~~~\begin{cases} {S}''_{i} (x_{i})=\dfrac{1}{h^{2}}({f}''_{0} (0)y_{i1} +{f}''_{1} (0)y_{i} +{f}''_{2} (0)y_{i+1} \\ \qquad\quad~~~+{f}''_{3} (0)y_{i+2} )=\dfrac{\pi ^{2}}{16h^{2}}(\alpha \beta +3)\\ \qquad\quad~~~~(y_{i1} 2y_{i} +y_{i+1}) \\[3mm] {S}''_{i} (x_{i+1})=\dfrac{1}{h^{2}}({f}''_{0} (1)y_{i1} +{f}''_{1} (1)y_{i} +{f}''_{2} (1)y_{i+1} \\ \qquad\qquad~~~+{f}''_{3} (1)y_{i+2} )=\dfrac{\pi ^{2}}{16h^{2}}(\alpha \beta +3)\\ \qquad\qquad~~~~(y_{i} 2y_{i+1} +y_{i+2}). \end{cases} \end{align} $  (8) 
From (6)(8), we have
$ \begin{align} \label{eq9} &\begin{cases} S_{i} (x_{i+1})=y_{i+1} =S_{i+1} (x_{i+1}) \\[2mm] {S}'_{i} (x_{i+1})=\dfrac{\pi} {16h}(3\alpha \beta +3)(y_{i+2} y_{i} )\\[5pt] \qquad\qquad={S}'_{i+1} (x_{i+1}) \\[2mm] {S}''_{i} (x_{i+1})=\dfrac{\pi^{2}}{16h^{2}}(\alpha \beta +3)(y_{i} 2y_{i+1} +y_{i+2})\\[5pt] \qquad\qquad\, ={S}''_{i+1} (x_{i+1}) \\ \end{cases}, \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ i=1, 2, \ldots, n3. \end{align} $  (9) 
Equation (9) shows that the CTAIspline curves
From Theorem 1, if two auxiliary points
$ \begin{align} \label{eq10} \begin{cases} (x_{  1} , y_{  1} ) = 2(x_0 , y_0 )  (x_1 , y_1 ) \\[1mm] (x_{n + 1} , y_{n + 1} ) = 2(x_n , y_n )  (x_{n  1} , y_{n  1} ). \end{cases} \end{align} $  (10) 
It is clear that there exist two degree of freedom in the CTAIspline curves, even if the data points and auxiliary points are fixed. Different shapes of the interpolation curves can be obtained by altering the parameters
Example 1: Given data points
$ \begin{align*} &(x_{0}, y_{0})=(1, 2), ~ (x_{1}, y_{1})=(0, 1)\\[1mm] &(x_{2}, y_{2})=(1, 3), ~ (x_{3}, y_{3})=(2, 4), ~ (x_{4}, y_{4})=(3, 2) \end{align*} $ 
and two auxiliary points
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Fig. 2 Different shapes of the 
Example 2: Given a function
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Fig. 3 Different shapes of the 
Example 3: Given proper data points and added two auxiliary points, some special curves generated by the
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Fig. 4 The notelike curves generated by the CTAIspline curves. 
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Fig. 5 The vanelike curves generated by the CTAIspline curves. 
Remark 1: When the traditional cubic polynomial splines are used to construct
Remark 2: On one hand, compared with some trigonometric spline curves (such as [6][8]), the main characteristic of the CTAIspline curves is that they can automatically interpolate the given data points without solving systems of equations. On the other hand, compared with the two trigonometric automatic interpolation splines presented in [11] and [12], the CTAIspline has the following characteristics,
a) The shape of the trigonometric automatic interpolation spline presented in [11] is unique when the data points and auxiliary points are fixed, while the CTAIspline curves can achieve shape adjustment by parameters
b) Although the shape of the trigonometric automatic interpolation spline presented in [12] can be adjusted when the data points and auxiliary points are fixed, the interpolation curves are only
As mentioned above, suppose a function
Example 4: Given a function
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Fig. 6 Effects of the parameters on the 
It is obvious that the interpolation curves in Fig. 3(a) are more satisfactory than the curves in Fig. 3(b). Hence, how to determine proper values of the parameters
Let
$ \begin{align} \label{eq11} F(\alpha, \beta)=\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {\left( {S_{i} (x)r(x)} \right)^{2}} {d}x} \end{align} $  (11) 
where
In order to obtain the minimum interpolation error, we have
$ \begin{align} \label{eq12} \begin{cases} \dfrac{\partial F(\alpha, \beta)}{\partial \alpha} =0 \\[3mm] \dfrac{\partial F(\alpha, \beta)}{\partial \beta} =0 .\\ \end{cases} \end{align} $  (12) 
Set
$ \begin{align*} &M_{01} (x)=\frac{1}{24}({14+9S+24S^{2}19S^{3}+14C^{3}} )\\[1mm] &M_{02} (x)=\frac{1}{24}({2+3SS^{3}+2C^{3}} )\\[1mm] &M_{03} (x)=\frac{1}{24}({69S+3S^{3}6C^{3}} )\\[1mm] &M_{11} (x)=\frac{1}{24}({29C2S^{3}+7C^{3}} )\\[1mm] &M_{12} (x)=\frac{1}{24}({103C+24S^{2}14S^{3}+13C^{3}} )\\[1mm] &M_{13} (x)=\frac{1}{24}({6+9C6S^{3}+9C^{3}} )\\[1mm] &M_{21} (x)=\frac{1}{24}({29S+7S^{3}2C^{3}} )\\[1mm] &M_{22} (x)=\frac{1}{24}({143S24S^{2}+13S^{3}14C^{3}} )\\[1mm] &M_{23} (x)=\frac{1}{24}({6+9S+9S^{3}6C^{3}} )\\[1mm] &M_{31} (x)=\frac{1}{24}({10+9C24S^{2}+14S^{3}19C^{3}} )\\[1mm] &M_{32} (x)=\frac{1}{24}({2+3C+2S^{3}C^{3}} )\\[1mm] &M_{33} (x)=\frac{1}{24}({69C6S^{3}+3C^{3}} ) \end{align*} $ 
where
Then, (5) can be rewritten as follows:
$ \begin{align} \label{eq13} S_{i} (x)=L_{1} (x)\alpha +L_{2} (x)\beta +L_{3} (x) \end{align} $  (13) 
where
$ \begin{align*} L_{1} (x)=&\ M_{01} (x)y_{i1} +M_{11} (x)y_{i} +M_{21} (x)y_{i+1} \\ &\, +M_{31} (x)y_{i+2}\\[1mm] L_{2} (x)=&\ M_{02} (x)y_{i1} +M_{12} (x)y_{i} +M_{22} (x)y_{i+1} \\ &\, +M_{32} (x)y_{i+2}\\[1mm] L_{3} (x)=&\ M_{03} (x)y_{i1} +M_{13} (x)y_{i} +M_{23} (x)y_{i+1} \\ &\, +M_{33} (x)y_{i+2}. \end{align*} $ 
By (13), (11) can be rewritten as follows:
$ \begin{align} \label{eq14} F(\alpha, \beta)=&\ \sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {S_{i}^{2} (x)} {d}x} 2\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1} } {S_{i} (x)} r{\rm (}x{)d}x} \nonumber\\ &\ +\sum\limits_{i=0}^{n1} {\int_{x_{i} }^{x_{i+1}} {r^{2}(x)} {d}x}\nonumber\\[2mm] =&\ C_{1} \alpha^{2}+C_{2} \beta^{2}+2C_{3} \alpha \beta +2C_{4} \alpha +2C_{5} \beta +C_{6} \end{align} $  (14) 
where
$ \begin{align*} &C_{1} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {L_{1}^{2} (x)} {d}x}\\ &C_{2} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {L_{2}^{2} (x)} {d}x}\\ &C_{3} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {L_{1} (x)L_{2} (x)} {d}x}\\ &C_{4} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {L_{1} (x)\left( {L_{3} (x)r(x)} \right)} {d}x}\\ &C_{5} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {L_{2} (x)\left( {L_{3} (x)r(x)} \right)} {d}x}\\ &C_{6} =\sum\limits_{i=0}^{n1} {\int_{x_{i}} ^{x_{i+1}} {\left({L_{3} (x)r(x)} \right)^{2}} {d}x}. \end{align*} $ 
From (14), (12) can be rewritten as follows:
$ \begin{align} \label{eq15} \begin{cases} C_{1} \alpha +C_{3} \beta +C_{4} =0 \\[1mm] C_{3} \alpha +C_{2} \beta +C_{5} =0. \end{cases} \end{align} $  (15) 
If
$ \begin{align} \label{eq16} \begin{cases} \alpha =\dfrac{C_{3} C_{5} C_{2} C_{4}} {C_{1} C_{2} C_{3}^{2}} \\[4mm] \beta =\dfrac{C_{3} C_{4} C_{1} C_{5}} {C_{1} C_{2} C_{3}^{2}}. \end{cases} \end{align} $  (16) 
Remark 3: If
After the optimal parameters
Since the trigonometric automatic interpolation spline curves presented by [11] are
Example 5: Given a function
The curve of the given function (marked with solid lines), the optimal
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Fig. 7 Interpolation results of Example 5 ( 
Example 6: Given a function
The curve of the given function (marked with solid lines), the optimal
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Fig. 8 Interpolation results of Example 6 ( 
Example 5 and Example 6 show that the interpolation result of the optimal
A class of cubic trigonometric automatic interpolation spline curves with two parameters is presented in this paper. The spline curves can not only automatically interpolate the given data points and be
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