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# 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 and let their centers be located a distance apart. Further suppose that a rod of length is fixed at each end to the circumference of the two wheels. Let be the midpoint of the rod. Then Watt's curve is the locus of .

The polar equation of Watt's curve is

 (1)

The areas of one of the inner lenses, heart-shaped half-region, and entire enclosed region (which resembles a lemniscate are

 (2) (3) (4)

If , then is a circle of radius with a figure of eight inside it.

Heart Curve, Nephroid, Watt's Parallelogram

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## 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.

Watt's Curve

## Cite this as:

Weisstein, Eric W. "Watt's Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WattsCurve.html