The hyperboloid is given by $\displaystyle H=\{(x,y,z) \vert x^2 + y^2 - z^2 =1\}$. Find coordinate patches that cover all of H, using the parametrization $\displaystyle \textbf{H}(s,t)=(\cos s \, \sec t ,\, \sin s \, \sec t ,\, \tan t)$.

First thing--we haven't covered "coordinate patch" in lecture, and the phrase does not occur anywhere in our text (incredible, but true--to be fair the lectures only cover the same general topics as the text, but notation and approaches are completely different). I presume that what I'm being asked is to find intervals for the parameters "s" and "t" such that the given parametrization gives all of the hyperboloid. Do I infer correctly?

Second, here is a proposed solution that I think is not quite right, but I can find no information to shoot it down directly with: Let $\displaystyle a < t < b$ where the length of the interval $\displaystyle (a,b)$ covers at least a full cycle of both $\displaystyle \sec t$ and $\displaystyle \tan t$. Similarly, let the other parameter meander over an interval covering both the sine and cosine functions.

What does not sound so right about this solution is that, for example, what happens if the interval that either parameter ranges over was $\displaystyle (0,4\pi)$? This certainly covers at least one full cycle of the trig functions here, but seems like it "does too much". I dunno, maybe this is okay...but maybe not. Seems like what I should be looking for is something of a "minimal" covering (why buy stuff you don't necessarily need, yea?). Here I'm at a total loss, though, because secant and tangent have different periods so wouldn't I be repeating some things anyway? Am I even in the right zip code?