# Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and $d_i$ are determined with the following constraints : $$\begin{array}{|c|c|c|c|} \hline &\varphi_{i,j}(x_i)&\varphi_{i,j}(x_{i+1})&\varphi_{i,j}^\prime(x_i)&\varphi_{i,j}^\prime(x_{i+1})\\ \hline j=0&1&0&0&0\\ \hline j=1&0&1&0&0\\ \hline j=2&0&0&1&0\\ \hline j=3&0&0&0&1\\ \hline \end{array}$$ I need to determinate the minimum tension factor $\tau$, on each interval. I've read this paper, but I don't really understood. Does anyone ever used that kind of interpolation ?

Ps. I apologize for my English.

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The idea with tension splines is that you keep adjusting those $\tau$ values up until you're satisfied with the appearance of your spline... –  Ｊ. Ｍ. Jul 9 '12 at 12:45
In the paper I read, there is a description of a method for select automatically the tension factor. –  Alexandre Hoffmann Jul 9 '12 at 13:08