<s>[INST] How level sets associated with a quadratic expression can switch from one to two components? So I have two equations:
$$
x^2 + y^2 - z^2 = 1/2
$$
and
$$
x^2 + y^2 - z^2 = -1/2,
$$
which represent single-sheeted hyperboloid, and a two-sheeted hyperboloid, respectively. How do you explain how this different sign has this effect on the level surfaces?
 [/INST] Note that the expression $x^2+y^2-z^2$ has rotational symmetry about the $z$-axis, which can be seen by introducing polar coordinates: 


*

*$x=r\cos t, y= r\sin t \Rightarrow r^2-z^2 = \pm\frac{1}{2}$
This means, any intersection of a plane containing the $z$-axis with $x^2+y^2-z^2 = \pm \frac{1}{2}$ produces the curve $r^2-z^2 = \pm\frac{1}{2}$ with $r\geq 0$.
Factoring shows that we have in both cases a hyperbolic curve:


*

*$(r-z)(r+z) = \pm\frac{1}{2} \Leftrightarrow r+z = \pm\frac{1}{2(r-z)}$
Now, just check the shape of the two curves (for example plotting on an $r$-$z$-coordinate system):


*

*$r^2-z^2 =\frac{1}{2} \Leftrightarrow z = \pm\sqrt{r^2-\frac{1}{2}} \Rightarrow$ only one branch (the seeming two branches are connected at $z =0 \Leftrightarrow r=\frac{1}{\sqrt{2}}$) $\Rightarrow$ one sheet

*$r^2-z^2 =-\frac{1}{2} \Leftrightarrow z = \pm\sqrt{r^2+\frac{1}{2}} \Rightarrow$ two branches $\Rightarrow$ two sheets
</s>