# Curve and admissible change of variable

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## Homework Statement

If I have the two curves

##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]##
##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ##

My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ## g^{-1} : [\frac{\pi}{4} , \frac{5 \pi}{4} ] \rightarrow [0, 2\pi]## and ##g^{-1}(s)= \frac{5\pi}{2} -2s ## (so ## g^{-1}(5\pi /4) = 0## and ##g^{-1}(\pi /4) = 2\pi ## ).

I found instead ##g^{-1}(s)= 2s - \frac{\pi}{2} ## ( so ##g^{-1}(\pi /4) =0## and ##g^{-1}(5 \pi /4) = 2\pi## )

Now the problem is, there are two admissible changes of variable and if I choose one or the other the way I go on the curve is opposite (because my ##(g^{-1})' > 0 ## and my textbook ## (g^{-1})' <0 ## ).

Now I don't understand if a change of variable can change the way of the curve.
Any help?