# How to find the range of a curve of intersection

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?

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 $$2\pi$$.