## 2 Proposed Spline Method  
Consider the data points \(\{(t_{i},F_{i}),\ i=0,1,2,\ldots,n\}\), where domain \(t_{i},\ i=0,1,2,\ldots,n\), represents knots with \(t_{0}<t_{1}<\cdots<t_{n}\) and \(F_{i},\ i=0,1,2,\ldots n\), be values at these knots. A rational quadratic trigonometric function \(R(t)\), with two shape parameters \(\rho_{i},\ \sigma_{i}>0,\ i=0,\)\(1,\ldots,\)\(n-1\), is defined over the interval \(I_{i}=[t_{i},t_{i+1}],\ i=0,1,2,\ldots,n-1,\ \text{as}\):  
\[R(t)=\sum_{i=0}^{3}S_{i}(t)P_{i}\,\ t\in[t_{i},t_{i+1}) \tag{1}\]  
\(S_{i}(t),\ i=0,1,2,3\) are rational quadratic trigonometric basis functions defined as:  
\[S_{0}(t)=\frac{A_{0}(t)}{S(t)},\ S_{1}(t)=\frac{\rho_{i}A_{1}(t)}{S(t)},\ S_{2}(t)= \frac{\sigma_{i}A_{2}(t)}{S(t)},\ S_{3}(t)=\frac{A_{3}(t)}{S(t)},\]  
where  
\[ \begin{array}{l}A_{0}(t)=(1-sin\theta)^{2},\ A_{1}(t)=sin\theta(1-sin\theta ),\\ A_{2}(t)=cos\theta(1-cos\theta),\ A_{3}(t)=(1-cos\theta)^{2}\end{array} \]  
and  
\[S(t)=A_{0}(t)+\rho_{i}A_{1}(t)+\sigma_{i}A_{2}(t)+A_{3}(t).\]  
Additionally,  
\[\theta(t)=\frac{\pi(t-t_{i})}{2h_{i}},\ 0\leq\theta\leq\frac{\pi}{2}\ \text{and}\ \ h_{i}=t_{i+1}-t_{i},\ i=0,1,\ldots,n-1.\]  
Furthermore, \(\sum_{i=0}^{3}S_{i}(t)=1\) as demonstrated in Figure 1. By applying the following \(C^{1}\)-continuity conditions  
\[ \begin{array}{l}R(t_{i})=F_{i},\ R(t_{i+1})=F_{i+1},\\ R^{(1)}(t_{i})=M_{i},\ R^{(1)}(t_{i+1})=M_{i+1},\end{array} \]  
Figure 1: Blending Functions. at the end points of the interval \(I_{i}=[t_{i},t_{i+1}]\), Bezier-like rational quadratic trigonometric Function (1) is transformed into a spline:  
\[R(t)=R_{i}(t;\rho_{i},\sigma_{i})=\frac{A_{0}(t)P_{0}+\rho_{i}A_{1}(t)P_{1}+ \sigma_{i}A_{2}(t)P_{2}+A_{3}(t)P_{3}}{S(t)}\]  
with  
\[P_{0}=F_{i},\ P_{1}=V_{i}=F_{i}+\frac{2h_{i}M_{i}}{\rho_{i}\pi},\ P_{2}=W_{i}=F_ {i+1}-\frac{2h_{i}M_{i+1}}{\sigma_{i}\pi},\ P_{3}=F_{i+1}. \tag{2}\]  
\(M_{i}\)'s are the computed values of 1 st derivatives at the knots \(t_{i}\). \(\rho_{i},\ \sigma_{i}>0\) provides positive denominator. For \(\rho_{i}=\sigma_{i}=2\), (2) turns into quadratic trigonometric interpolant \(R(t)\in C^{1}[t_{1},t_{n}]\). Now by putting the following \(C^{2}\) continuity conditions  
\[R_{i-1}^{(2)}(t_{i-})\Big{|}_{\theta_{i}=\frac{\pi}{2}}=R_{i}^{(2)}\big{(}t_{ i}^{+}\big{)}\Big{|}_{\theta_{i}=0},\ i=1,2,\ldots n-1. \tag{3}\]  
The following tri-diagonal system of linear equations in \(n-1\) unknowns \(M_{i}\), \(i=1\), \(2,\ldots,n-1\), is achieved as follows:  
\[h_{i}M_{i-1}+2\big{\{}(\sigma_{i-1}-1)h_{i}+(\rho_{i}-1)h_{i-1}\big{\}}M_{i}+ h_{i-1}M_{i+1}=\frac{\pi}{2}(\rho_{i-1}h_{i}\Delta_{i}+\sigma_{i}h_{i-1} \Delta_{i}), \tag{4}\]  
where  
\[\Delta_{i}=\frac{F_{i+1}-F_{i}}{h_{i}},\ i=1,\ 2,\ldots,n-1.\]  
With suitable end conditions, the above system is diagonally dominant and a unique solution for \(M_{i}\)'s can be computed with the following limits on the shape parameters,  
\[\rho_{i}\sigma_{i}=\alpha\geq 2,\ i=1,\ 2,\ldots n. \tag{5}\]  
The preceding discussion can be summarized as follows:  
**Theorem 1**. : _RQTS has a solution that is unique due to the restriction on the shape parameters (5)._  
**3. Error Analysis of RQTS**  
The error of RQTS Function (1) is calculated in this segment by the following theorem. **Theorem 2**. : _For \(G(t)\in C^{3}[t_{0},t_{n}]\), let \(R(t)\) be the rational quadratic trigonometric spline (1) interpolates \(G(t)\) in \([t_{0},t_{1}]\), then for \(\zeta_{i}>0,\ \xi_{i}>0\), the following holds:_  
\[|G(t)-R(t)|\leq\|G^{3}(\delta)\|h_{i}^{3}b_{i},\]  
_where_  
\[b_{i}=\max_{0\leq\sigma\leq 1}\chi(\zeta_{i},\xi_{i},\theta),\]  
_with_  
\[\chi(\zeta_{i},\xi_{i},\theta)=\left\{ \begin{array}{c}\max\chi_{1}(\zeta_{i},\xi_{i},\theta),\ 0\leq\xi_{i}\leq 1,\ 0\leq\sigma\leq 1\\ \max\chi_{2}(\zeta_{i},\xi_{i},\theta),\xi_{i}>1+\frac{\pi}{2},\ 0\leq\sigma\leq\sigma^{*}\\ \max\chi_{3}(\zeta_{i},\xi_{i},\theta),\ \xi_{i}>1+\frac{\pi}{2},\ \sigma^{*}\leq \sigma\leq 1.\end{array}