4.5: The Hyperboloid
The equation
\[\frac{x^2}{a^2} - \frac{z^2}{c^2} = 1 \label{4.5.1} \]
is a hyperbola, and \(a\) is the semi transverse axis. (As described in Chapter 2, \(c\) is the semi transverse axis of the conjugate hyperbola.)
If this figure is rotated about the \(z\)-axis through \(360^\circ\) , the surface swept out is a circular hyperboloid (or hyperboloid of revolution) of one sheet. Its equation is
\[\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1. \label{4.5.2} \]
Imagine two horizontal rings, one underneath the other. The upper one is fixed. The lower one is suspended from the upper one by a large number of vertical strings attached to points equally spaced around the circumference of each ring. Now twist the lower one through a few degrees about a vertical axis, so that the strings are no longer quite vertical, and the lower ring rises slightly. These strings are generators of a circular hyperboloid of one sheet.
If the figure is rotated about the \(x\)-axis through \(360^\circ\) , the surface swept out is a circular hyperboloid (or hyperboloid of revolution) of two sheets. Its equation is
\[\frac{x^2}{a^2} - \frac{y^2}{c^2} - \frac{z^2}{c^2} = 1. \label{4.5.3} \]
The equations
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \label{4.5.4} \]
and \[\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \label{4.5.5} \]
represent hyperbolas of one and two sheets respectively, but are not hyperbolas of revolution, since their cross sections in the planes \(z =\) constant and \(x =\) constant \(> a\) respectively are ellipses rather than circles. The reader should imagine what the cross- sections of all four hyperboloids are like in the planes \(x = 0, \ y = 0\) and \(z = 0\).