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In the following parameterization of a cone: $$r(u,\theta) = u \cos v i + u \sin \theta j + u k, 0 \le u \le r$$ for a constant $r$, my understanding is that $\theta$ represents the angle that "goes around" the circle at a height $u$ from the origin.

However, in Problem 12.10 8 at this link, the bounds that are put on $\theta$ are from $- \frac{\pi}2$ to $\frac{\pi}2$. Why are they not from $0$ to $2 \pi$ to complete the circle?

Euclid
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1 Answers1

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Note that the cylinder only intersects the cone in places with nonnegative $x$-coordinate; we therefore only only need to use $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ as this is the set of angles with such coordinate.

Put another way, $0\le u\le 2\cos\theta$, and thus $\cos\theta\ge0$ which means $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

auscrypt
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