Watt's curve

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Description:
In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate. It arose in connection with James Watt's pioneering work on the steam engine.

The equation of the curve can be given in polar coordinates as
<math>r^2=b^2-left^2.</math>


Derivation

Polar coordinates

The polar equation for the curve can be derived as follows:Working in the complex plane, let the centers of the circles be at a and −a, and the connecting segment have endpoints at −a+be<sup>i λ</sup> and a+be<sup>i ρ</sup>. Let the angle of inclination of the segment be ψ with its midpoint at re<sup>i θ</sup>. Then the endpoints are also given by re<sup>i θ</sup> ± ce<sup>i ψ</sup>. Setting expressions for the same points equal to each other gives
<math>a+be^=re^+ce^.,</math>
<math>-a+be^=re^-ce^,</math>
Add these and divide by two to get
<math>re^=tfrac(e^+e^)=bcos(tfrac)e^.</math>
Comparing radii and arguments gives
<math>r=bcosalpha, theta=tfrac mbox alpha=tfrac.</math>
Similarly, subtracting the first two equations and dividing by 2...
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