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Give an explicit example of:

$(a)$ a regular curve parametrized by arclength;

$(b)$ a non-regular curve which has the same geometric image as the previous one.

Could someone please help me with this? I think I am missing a definition that would make this simpler.

I am confused as what it means for two curves to have the same geometric image. I think that the image is the derivative of the parametrized curve, is this correct? If so, would this work:

$a)$ $\alpha(s) = (r\cos(s/r), r\sin(s/r), 0)$

$b)$ $\alpha(s) = (r\cos(s/r-\pi/2), r\sin(s/r), 0)$

Thank you in advance!

Edit: My example would not work because they don't have the same derivative, but do they still have the same image?

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Well, consider the following two parametrizations of the unit circle :

  • $c(s):=(\cos s,\sin s,0),\ \ s\in \mathbb{R}$ and
  • $\tilde{c}(t):=(\cos (t^2),\sin (t^2),0), \ \ t\in \mathbb{R}$

Clearly the first parametrization is by arclength and the second is a non-regular parametrization since $$\tilde{c}'(0)=(0,0,0).$$

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  • $\begingroup$ Thank you very much! I hadnt thought of that! $\endgroup$ Feb 12 '16 at 15:37

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