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?