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I have a set of data. (when plotted looks like following )

plot of user-provided data as example

I need to find out the x and y coordinates of the points where the common tangent touches the curve. Which free software on Ubuntu can help me to get the coordinates of the points where the common tangent touches the curve ?

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I'm not sure exactly what you are looking for but if you want something like Linux equivalent to Matlab then try GNU Octave.

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A mathematica-like environment can be found here : http://www.mathics.org/

There is an online version, and you can install it (install instructions on the site). This can surely help you, although you need to know the math obviously :)

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