Watt's Curve

Polar equation:
r2 = b2 - [a sin(θ) plusminus √(c2 - a2cos2(θ))]2

Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

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The curve is named after James Watt (1736- 1819), the Scottish engineer who developed the steam engine. In fact the curve comes from the linkages of rods connecting two wheels of equal diameter.

Let two wheels of radius b have their centres 2a apart. Suppose that a rod of length 2c is fixed at each end to the circumference of the two wheels. Let P be the mid-point of the rod. Then Watt's curve C is the locus of P.

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

Sylvester, Kempe and Cayley further developed the geometry associated with the theory of linkages in the 1870's. In fact Kempe proved that every finite segment of an algebraic curve can be generated can be generated by a linkage in this way.

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JOC/EFR/BS January 1997

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