What is the plot of this function?

Question

$$\mathrm{f}: \mathbb{R}^{2} \to \mathbb{R}^{3} : \left(x,y\right) \to \left(\cosh\left(x\right)\cos\left(y\right),\cosh\left(x\right)\sin\left(y\right), \sinh\left(x\right)\right) $$

Answer 1

If $X(x,y,z) = \cosh x \cos y$, $Y(x,y,z) = \cosh x \sin y$, $Z(x,y,z) = \sinh x$, then $X^2 + Y^2 - Z^2 = 1$ (because we have the hyperbolic trigonometric identity $\cosh^2 - \sinh^2 = 1$), and that equation is known to describe points living on a 1-sheet hyperboloid having $z$ for axis of symmetry and having circles for the cross-sections parallel to the plane $xOy$.

It is not difficult to show that your formula covers all the points of this hyperboloid: $x$ running across $\Bbb R$ makes your points reach any height, and for fixed $x$, letting $y$ run across $\Bbb R$ makes your points turn around the symmetry axis, at height $\sinh x$.

Your formula is not a parametrization, though, because it covers each point of the hyperboloid infinitely many times; it is only a covering map. In order to get a parametrization, you should restrict your domain of definition to something like $\Bbb R \times [0, 2\pi)$. If you also want to speak about differentiability, then you should consider $(0, 2\pi)$ (notice the open end at the left).

Answer 2

In cylindrical coordinates,

$$\theta=y,\rho=\cosh y,z=\sinh y$$ and $$\rho^2-z^2=1$$ which is the equation of an equilateral hyperbola.

Interestingly, this is a ruled surface, meaning that it can be generated from straight lines.