Calculus: Analytic Geometry and Calculus, with Vectors

6.6 Quadric surfaces 405

Figure 6.641

The graph of the equation

(6.65) a2 --b2-z2=-1 or

Figure 6.642

x2 y2 z2

-a2-b2+c2=1

is a hyperboloid of two sheets. It intersects the plane having the equation
Figure 6.651 z = k in the empty set whenIkj < c, in a point
when Iki = c, and in an ellipse (or circle) when
jkt > c. It intersects the planes having the
equations y = k and x = k in hyperbolas.
Unshaded and shaded graphs appear in Fig-
ures 6.651 and 6.652.
Figure 6.652