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Here's something really weird. As can be read in Pressley's "Elementary Differential Geometry":
Proposition 1.3: Any reparametrization of a regular curve is regular.
And 4 pages later:
Exemple 1.8: For the parametrization \gamma(t)=(t,t^2) of the parabola y=x², \dot{\gamma} is never 0 so \gamma is regular. But \tilde{\gamma}(t)=(t^3,t^6) is also a parametrization of the same parabila. This time, \dot{\tilde{\gamma}}=(3t^2,6t^5) and this is zero when t=0, so \tilde{\gamma} is not regular.
Just to make sure that \tilde{\gamma} is a reparametrization of \gamma, consider the reparametrization map \phi:(-\infty,+\infty)\rightarrow (-\infty,+\infty) define by \phi(t)=t^3. Then \phi is a smooth bijection with a smooth inverse such that \gamma \circ \phi = (\phi(t),\phi(t)^2)=(t^3,t^6)= \tilde{\gamma}, so \tilde{\gamma} is really a reparametrization of \gamma but it is not regular, contradicting proposition 1.3.
Proposition 1.3: Any reparametrization of a regular curve is regular.
And 4 pages later:
Exemple 1.8: For the parametrization \gamma(t)=(t,t^2) of the parabola y=x², \dot{\gamma} is never 0 so \gamma is regular. But \tilde{\gamma}(t)=(t^3,t^6) is also a parametrization of the same parabila. This time, \dot{\tilde{\gamma}}=(3t^2,6t^5) and this is zero when t=0, so \tilde{\gamma} is not regular.
Just to make sure that \tilde{\gamma} is a reparametrization of \gamma, consider the reparametrization map \phi:(-\infty,+\infty)\rightarrow (-\infty,+\infty) define by \phi(t)=t^3. Then \phi is a smooth bijection with a smooth inverse such that \gamma \circ \phi = (\phi(t),\phi(t)^2)=(t^3,t^6)= \tilde{\gamma}, so \tilde{\gamma} is really a reparametrization of \gamma but it is not regular, contradicting proposition 1.3.
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