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Physics LibreTexts

4.5: The Hyperboloid

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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.


This page titled 4.5: The Hyperboloid is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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