Watt's Curve

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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 2*a* apart (taken to be at (±*a*, 0). A line segment of length 2*c* 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

## 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

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The equation of the curve can be given in polar coordinates as

- <math>r^2=b^2-left^2.</math>

- <math>a+be^=re^+ce^.,</math>
- <math>-a+be^=re^-ce^,</math>

- <math>re^=tfrac(e^+e^)=bcos(tfrac)e^.</math>

- <math>r=bcosalpha, theta=tfrac mbox alpha=tfrac.</math>

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