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I have started reading a book on the subject and the author writes:

"one should carefully distinguish a parametrized curve, which is a map, from its trace, which is a subset of $\mathbb{R}^3$"

What does he mean?

If we look at $\alpha(t)=cost,sint$ the map is the function whereas the trace is a "graph" in a the shape of a circle?

gbox
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    The key point is that the same trace can be traversed with different velocities. – Miguel Apr 24 '17 at 13:52
  • @MiguelAtencia for example in my example a "faster" trace will be $a(t)=cos(5t),sin(5t)$? – gbox Apr 24 '17 at 13:53
  • The answer to your last question is "yes". Think of a car going around a track. The position of the car as a function of the parameter time is the parameterized curve. The trace is the track. – Ethan Bolker Apr 24 '17 at 13:55
  • A map gives you explicit directions on how to traverse the trace: you lose information by forgetting the map and just remembering the trace. – Joppy Apr 24 '17 at 13:55
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    @gbox But this change of parametrization can be "ill". For instance, $\alpha(t)=(\cos t^2,\sin t^2)$ gives null spped at $t=0$ so a singular point arises. https://en.wikipedia.org/wiki/Singular_point_of_a_curve#Parametric_curves – Miguel Apr 24 '17 at 13:59

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