There are two quadrics (surfaces defined by a quadratic function of three variables -- x, y and z) known as hyperboloids.
The hyperboloids are defined by the equation
z2 - x2 - y2 = C
If C is a positive quantity we get the hyperboloid of two sheets. If C is negative we get the hyperboloid of one sheet. The intermediate case (when C=0) produces a degenerate hyperboloid -- a double cone.
The animation below shows a sequence of images of the surfaces defined by
z2 - x2 - y2 = C
as C varies from -4 to +4. Thus we see a sequence of images wherein a hyperboloid of one sheet morphs into a hyperboloid of two sheets, momentarily passing through the "double cone" stage.
