Just parameterizing by a general parameter $t$ I got $\mathbf{r}(t) = \langle \cos t, \sin t, 2-\sin t\rangle$, which in the arc length formula would be: $$\mathbf{s}(t) = \int_0^t \sqrt{(-\sin u)^2 + \cos^2u + (-\cos u)^2}\ du = \int_0^t \sqrt{1+\cos^2u}\, du= \, ?$$ Am I doing something wrong, or is there just no way to parameterize by arc length?
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You forgot the square root (and the $du$) in the second expression of the integral. – Marc-André Brochu May 14 '23 at 02:51
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There's no closed form to parametrized an ellipse by its arclength. See another post for your interest. – Ng Chung Tak May 14 '23 at 06:57