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For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) \end {align}$$ How do we show that the two parametric equations draw the same line?

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    What if the parametric equations are all rational functions? In this case, it's often possible to implicitize -- i.e. convert to equations of the form $f(x,y)=0$ and $g(x,y)=0$. Then, if the two curves are the same point set, I would guess that something can be said about $f$ and $g$? maybe one is a multiple of the other?? Need comments from someone who knows more about algebraic geometry than I do.2013-04-28
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    Maybe I should turn my comment above into a separate question, so as not to divert this. So, that's what I did.2013-04-28

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