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Is there a way to produce parametric equations for a curve?(If we do know cartesian coordinates of course)

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What do you know about the curve then? You could always take a piece of rope and measure the length of the curve along piece wise arcs and thus parametrize it discretely. –  Raskolnikov Jan 16 '11 at 10:45
    
My mistake when i wrote the question.I corrected it –  t.k Jan 16 '11 at 12:32

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See Chapter 4 of this (a textbook on differential geometry). Essentially, a curve $\textbf{r}$ depends on two parameters ($u^1$ and $u^2$) in some arbitrary Cartesian coordinate system.

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I recommend the online book, intuitive introduction, for more deep stuff read the other reply. They call it multivariable calculus, and it covers some parametrization of curves. Sorry but not sure what you are asking here so just giving you a book to look at.

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Yes, you fit a cubic-spline through the points. Actually two cubic splines, one for $x$ and one for $y$ in terms of some independent parameter. You can make the parameter $t=0\cdots1$ or whatever you want.

In camshaft profile development we do that all the time, as we take measured points and fit splines to parametrize the curve and then calculate velocity, acceleration, jerk, snap, crackle and pop (1st,2nd,3rd,4th,5th,6th derivatives). Some smoothing is required sometimes.

There are free online books for cubic spline interpolation in C (link), and Fortran (link) which is easier to read and re-code to something more modern. A Fortran 90 version uses vectorized operations which are only useful in Matlab, or Fortran. I recommend reading chapter 3-3.

I have ported this routines in C# if interested.

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