where both a and b > 0.
*James
Watt* (1736-1819) was the guy who developed the steam engine. From the
movement of the wheels of the locomotives he derived the **Watt's curve**.
The curve describes the path of the midpoints of a rotating rod: given two wheels
(e.g. of a steam locomotive) of radius R (AD=BC in picture), having their centers a distance
d (CD) from each other. A rod of length l (AB) is fixed at each end to the circumference of the two
rotating wheels. Then the mid-point (M) of the rod describes the path of a Watt's
curve ^{1}.
The curve is a tricircular algebraic curve.
The octic Cartesian equation following from the polar equation above ^{2)}^{ }can
we simplified to a sextic
equation ^{3)}.
When the length of the rod and the distance between the circle's centers are
equal (i.e. b=1), the curve is a hippopede.
In this case, for parameter 'a' having the value 1/2 (the length of the rod is √2 times
the radius of the circle) a lemniscate
occurs (surrounded by a circle).
A Watt's curve describes a linkage. Mathematicians like *Sylvester*, *Kempe*
and *Cayley* worked on the theory of linkages in the 1870's. It can be
proved that every finite segment of an algebraic curve can be generated by a linkage.
**
****notes**
1) Parameters a and b can be expressed in the given distances R, d and l as follows:
- a = d
^{2} / 4 R^{2}
- b = l
^{2} / d^{2}
2) In Cartesian coordinates: 4y^{2}((x^{2}+y^{2})b^{2}-x^{2})=((x^{2}+y^{2})(x^{2}+y^{2}-a^{2}+b^{2})-x^{2}+y^{2})^{2
}
3) Sextic Cartesian equation: (x^{2}+y^{2})^{3}-2(x^{2}+y^{2})^{2}+(x+4y^{2})(x^{2}+y^{2})=dy^{2} |