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- Are two maps with the same image actually considered two different curves ?

Hi,

I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##.

Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the (common) image the two maps can have different velocities (i.e. different tangent vectors).

The question is: are they actually two different curves ?

Thank you.

I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##.

Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the (common) image the two maps can have different velocities (i.e. different tangent vectors).

The question is: are they actually two different curves ?

Thank you.