Parametric equations for the surface of revolution generated by rotating the hyperbola $z^2−y^2=2$ about the $y$-axis

Question

Write the parametric equations of the surface of revolution generated by the hyperbola $$\frac{z^{2}}{2} − \frac{y^{2}}{2} = 1$$ (in the $yz$ -plane) when it rotates around the $y$-axis.

Any hint in how to start?

I guess it $y$ coordinate of the parametric equation is simply $y = y$. For $z$ and $w$ it will be a radius multiplied by $\cos\theta$, $\sin\theta$ respectively. But I have no idea how to obtain such radius.

Answer 1

Hint We can parameterize (one component of) the curve $$u^2 - v^2 = 1$$ via $$u(t) = \cosh t, \qquad v(t) = \sinh t .$$