Family of Hyperboloids
This movie shows the sequence of level surfaces obtained from
x2 + y2 - z2 = kfor values of k from -1 to 1. For k < 0, the level surface is a hyperboloid of two sheets. As k gets closer to 0, the two sheets move closer together, and at k = 0, they meet, forming a double cone. For k > 0, the surface is a hyperboloid of one sheet.
The change from two sheets to one sheet is a significant change in shape of the level surface. The double cone that separates the two is a critical level for the function
f(x,y,y) = x2 + y2 - z2,and the origin is a critical point.
Note that these surfaces form a nested family of surfaces and that no two of them intersect. When "stacked up" in four-space, they form the graph of the function f. These are analogous to the level curves of the function
f(x,y) = x2 - y2,which are hyperbolas opening around the y axis when k < 0or around the x axis for k > 0. The transition occurs at k = 0, as two crossing lines, a critical level for this function. When "stacked up" in three-space, these form the saddle surface, with a critical point at the origin.