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

* z ^{2} - x^{2} - y^{2} = 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

* z ^{2} - x^{2} - y^{2} = 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.

fieldsj1@SouthernCT.edu