The simplest form for the equation of a sphere in $\mathbb{R}^3$ is:
$$x^2 + y^2 + z^2 = r^2$$
where $r$ is the sphere radius. This substantially is the Pythagorean Theorem, applied to the triangle whose squared sides are
- $a^2 = x^2 + y^2$
- $z^2$
$r^2$ is the square of the resulting hypotenuse.
Conversely, the simplest form for the equation of a hyperboloid is:
$$x^2 + y^2 - z^2 = k^2$$
where $k$ is a real fixed number.
If the equation of a sphere represents the Pythagorean Theorem, what does instead the equation of a hyperboloid represent? Is there any other intuitive idea behind it?
Due to the minus sign, it can't be the Pythagorean Theorem again, unless the length $z$ is considered as pure imaginary.
There is in fact a geometric/intuitive interpretation of a hyperboloid. For any hyperboloid, there exist points $A$ and $B$ (the foci of the hyperboloid) such that the hyperboloid is the locus of points satisfying $|AP - BP| = c$ for some constant $c$. You can simply see the hyperboloid as a rotation of a hyperbola around its axis. This also satisfies reflective properties analogous to the hyperbola. For more information, see here: https://en.wikipedia.org/wiki/Hyperbola