Family of Hyperboloids

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This movie shows the sequence of level surfaces obtained from x2 + y2 - z2 = k for 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 < 0 or 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.

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