Watt's Curve

Polar equation:
r2=b2[asin(θ)±(c2a2cos2(θ))]2r^{2} = b^{2} - [a \sin( \theta ) ± √(c^{2} - a^{2}\cos^{2}( \theta ))]^{2}


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 bb have their centres 2a2a apart. Suppose that a rod of length 2c2c is fixed at each end to the circumference of the two wheels. Let PP be the mid-point of the rod. Then Watt's curve CC is the locus of PP.

If a=ca = c then CC is a circle of radius bb 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.