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Suppose we want to find the length of the curve that is the intersection of the surfaces $x^2+y^2+z^2=1$ and $x^2+2z^2=1$.

One can parametrize the curve as

$$ r(t) = \left( \cos t, 1/\sqrt{2} \sin t, 1/\sqrt{2} \sin t \right) $$

So that the length is just $\int || r' || dt $ but, my question is, how do we find the out the starting and final point of the curve?

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The parametrisation consists of $2\pi$-periodic trigonometric functions, so $r$ will trace the curve precisely once for $t$ restricted to in an interval of length $2\pi$, e.g. $[0,2\pi]$.

An easy way to see this is by looking at the first component of $r$ which is $\cos$. Since $\cos(t)=1$ only if $t\in\{0,2\pi\}$ we know that $r(t)\ne r(0)$ unless $t\in\{0,2\pi\}$, so $r$ definitely doesn't trace the same curve before $2\pi$. And then note that $r(t)=r(t+2\pi)$ to confirm that $r$ does trace the same curve after $2\pi$.

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