# hyperboloid of two sheets

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## hyperboloid of two sheets

[hī′pər·bə‚lȯid əv ′tü ‚shēts]
(mathematics)
A surface whose equation in standard form is (x 2/ a 2) - (y 2/ b 2) - (z 2/ c 2) = 1, so that it is in two pieces, and cuts planes perpendicular to the y and z axes in hyperbolas and planes perpendicular to the x axis in ellipses, except for the interval -a <>x <>a, where there is no intersection.
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If the variable X' is used, the equations for the hyperboloid of two sheets becomes [(x'/a).sup.2] - [(y'/b).sup.2] - [(z'/c).sup.2] -1 = 0, where the semi axes for the hyperboloid are a [approximately equal to] 0.84365, b [approximately equal to] 1.31066, and c [approximately equal to] 0.77172, respectively.
In the case of the hyperboloid of two sheets, we observe that [k.sub.0] < 0 and we obtain two positive solutions of the quadric equation (45) which correspond to two different sheets of the hyperboloid.
In the case of a hyperboloid of two sheets, we use the change of variables equation (33), to get [??] with the components

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