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4.5: The Hyperboloid

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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 $$z = 0$$.