Watt's Curve

A curve named after James Watt (1736-1819), the Scottish engineer who developed the steam engine (MacTutor Archive). The curve is produced by a Linkage of rods connecting two wheels of equal diameter. Let the two wheels have Radius $b$ and let their centers be located a distance $2a$ apart. Further suppose that a rod of length $2c$ is fixed at each end to the Circumference of the two wheels. Let $P$ be the Midpoint of the rod. Then Watt's curve $C$ is the Locus of $P$.

The Polar equation of Watt's curve is

$$r^2=b^2-(a\sin\theta\pm\sqrt{c^2-a^2\cos^2\theta}\,)^2.$$

If $a = c$, then $C$ is a Circle of Radius $b$ with a figure of eight inside it.

References

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 162, 1967.

MacTutor History of Mathematics Archive. ``Watt's Curve.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Watts.html.