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Is there an algorithm or implementation? I have searched the monotone cubic interpolation or monotone piecewise cubic interpolation. It seems that both the two methods cannot preserve the derivatives at the two given points. I am not really sure about this, because I am not a mathematician. Can you provide some information or ideas? Anything will be greatly appreciated.

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What exactly do you want? Are you given two points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1<x_2$ and $y_1<y_2$, as well as numbers $a>0$, $b>0$ and you want a smooth function $f$ such that $f(x_1)=y_1$, $f(x_2)=y_2$, $f'(x_1)=a$, $f'(x_2)=b$, $f'(x)>0$ for all $x\in[a,b]$? – Hagen von Eitzen Feb 1 '13 at 15:55
yes, exactly what i want – user1446072 Feb 2 '13 at 4:51

I just found Bézier curve meet my requirements perfectly and is really simple for implementation.

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