The hyperboloid as a level surface

The picture below shows the level surface

$f^{-1}(k)$ of the function

$f : \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by

$f(x, y, z) = x^2 + y^2 - z^2$ .

You can move the slider on the left to change the value of

$k$ . Note that at

$k=0$ the surface becomes singular (it is a cone). When

$k > 0$ the surface is connected, and is called the Hyperboloid of one sheet. When

$k < 0$ the surface has two components, and is called the Hyperboloid of two sheets