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Watt's Curve

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WattsCurve

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-(asintheta+/-sqrt(c^2-a^2cos^2theta))^2.
(1)
WattsCurveAreas

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

A_(lens)=1/2pi(b^2-c^2)-asqrt(c^2-a^2)-c^2tan^(-1)(a/(sqrt(c^2-a^2)))
(2)
A_(heart)=pi(b^2-c^2)
(3)
A_(enclosed)=pi(b^2-c^2)+2asqrt(c^2-a^2)+2c^2tan^(-1)(a/(sqrt(c^2-a^2))).
(4)

If a=c, then C is a circle of radius b with a figure of eight inside it.

See also

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.

Referenced on Wolfram|Alpha

Watt's Curve

Cite this as:

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

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