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I have this problem:

I have a curve/figure on a sheet of graphpaper. Then I have to select points, read the x,y-values and label them $t_{0} = 1.0$ and $ t_{1}=2.0$ ect. I now have to obtain a table of $x(t)$ and a table of $y(t)$.Then I have to fit these functions by spline functions. Thus I have $x(t) = S(t)$ and $y(t) = S^{*}(t)$, which gives an approximate parametric representation of the curve/figure.

My question is:

Why should I have to fit them with two splines function? When in fact I just can fit the table of y(x) with one spline function?

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It depends on what your curve/figure looks like. If it doesn't satisfy the vertical-line test, then you do need to use a parametric representation... –  J. M. Oct 14 '11 at 8:43
    
What is the vertical line test? The curve is an oval or a spiral. –  user145 Oct 14 '11 at 8:45
    
See this. An oval would not pass the vertical line test, since a typical vertical line would intersect it at least twice (and yes, a tangency counts as intersecting twice); a spiral would intersect a vertical line quite a number of times. So yes, you do need a parametric representation in those cases. –  J. M. Oct 14 '11 at 8:49
    
Maybe your teacher is trying to get you to understand parametric representations. –  Peter Taylor Oct 14 '11 at 9:18

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