Family of Hyperboloids |
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This movie shows the sequence of level surfaces obtained from
x^{2} + y^{2} - z^{2} = k for values of k from -1 to 1. Fork < 0 , the level surface is a hyperboloid of two sheets. As k gets closer to 0, the two sheets move closer together, and atk = 0 , they meet, forming a double cone. Fork > 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) = x^{2} + y^{2} - z^{2}, 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) = x^{2} - y^{2}, which are hyperbolas opening around the y axis whenk < 0 or around the x axis fork > 0 . The transition occurs atk = 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.