Watt's curve

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 partially back and forth or completely around.

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 The polar equation for the curve can be derived as follows:[1] Working in the complex plane, let the centers of the circles be at a and −a, and the connecting segment have endpoints at −a+bei λ and a+bei ρ.

Let the angle of inclination of the segment be ψ with its midpoint at rei θ.

Setting expressions for the same points equal to each other gives Add these and divide by two to get Comparing radii and arguments gives Similarly, subtracting the first two equations and dividing by 2 gives Write Then Expanding the polar equation gives Letting d 2=a2+b2–c2 simplifies this to The construction requires a quadrilateral with sides 2a, b, 2c, b.

If b=a+c then two branches of the curve meet at the origin with a common vertical tangent, making it a quadruple point.

If a=c then the curve decomposes into a circle of radius b and an oval of Booth.

If a>c then the curve does not cross the x-axis at all and consists of two flattened ovals.