A quadric whose equation in a suitable coordinate system is

The axial planes are planes of symmetry. The section by a plane z = k parallel to the xy‐plane is, when non‐empty, an ellipse (a circle if a = b). When −c<k<c, the plane z = k has no points of intersection with the hyperboloid, and the hyperboloid is thus in two pieces or sheets. Sections by planes parallel to the other two axial planes are hyperbolas.

Hyperboloid of two sheets in normal form