I do not understand why when you revolve a hyperbola around a circle the respective parameters (cosh (v) and cos (u)) are multiplied by each other to get the parametric form of the hyperboloid. I would greatly appreciate it if someone could show me how these parameters are reached.
2026-03-31 13:42:39.1774964559
Parametrizaction of a Hyperboloid
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I'm guessing a bit at the description you're looking at, but I think you're looking at $$ (\cosh (v) \cdot \cos (u), \cosh (v) \cdot \sin (u), \sinh(v)) $$
Think of your hyperbola in the $x,z$ plane, and think about what happens when you rotate a single point on the hyperbola around the $z$-axis, say $$(\cosh(v),0,\sinh(v))$$ Well, it's a circle of radius $\cosh(v)$ maintaining a height of $\sinh(v)$. So the $z$ coordinate stays fixed, and the $x,y$ coordinates pass through $(\cos(u)\cdot\cosh(v),\sin(u)\cdot\sinh(v))$. I'm not great at describing these things without chalk, but does that make sense?