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I have a question where I am given 2 space curves:

$$r(t)=\langle\cos{t},\sin{t},t\rangle$$

$$q(s)=\langle\cos{4s},\sin{4s},s\rangle$$

where both $t$ and $s$ are between $0$ and $2\pi$ exclusive. I am supposed to find where these two lines intersect. So I set $\cos{t}$ equal to $\cos{4t}$ and tried to solved. However, I cannot seem to get an answer from doing this. Please tell me what I'm doing wrong and how I can correctly find the points of intersection (of which there are supposedly 2).

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1 Answer

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Note that in general when two parametric curves intersect, it doesn't have to be at the same value of the parameter for both.

Now in this case it turns out you are right, because of the third coordinate: $$t=s.$$

Now note that $$\cases{\cos t =\cos u\Longleftrightarrow t=\pm u\bmod 2\pi\\\sin t =\sin u\Longleftrightarrow [t= u\bmod 2\pi \ \ \text{ or }\ \ t= \pi-u\bmod 2\pi ]}$$ For any $t$, $u$, therefore including the case $u=4t$.

Can you take it from here?

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