The equation

$x^2 - y^2 + z^2 = 8$

represents a hyperboloid in three-dimensional space.

Step-by-Step Analysis:

1. Rewriting the Equation: $x^2 + z^2 - y^2 = 8$ This equation has the form of a hyperboloid of one sheet when written as: $\frac{x^2}{8} + \frac{z^2}{8} - \frac{y^2}{8} = 1.$ This is the standard form of a hyperboloid of one sheet along the $y$-axis.

2. Shape and Properties:

   • It is a saddle-shaped surface extending infinitely in the $x, y,$ and $z$ directions.
   • It has two asymptotic cone sections when viewed from afar.
   • Cross-sections:
     • At $y = 0$: The equation simplifies to $x^2 + z^2 = 8$, which is a circle of radius $\sqrt{8}$.
     • For other values of $y$, the cross-sections are ellipses.

Visualization:

Would you like me to generate a 3D visualization of this surface?