;;;r7;;;
Z2288887 ;8888aS0:
.822S ;aS0:
SZ0i .; ,082
;8S . XZX
X0 : 0S
88 .i aW
2a . Xa
.0 .i W:
2X . :0
W B
M .; M
B . 8,
M : @
W .i @
B . Xi
M .i M
2 . ;;
B, M
B .; B
B . @
0: : M
M XZ88aXSSSSX08882r . ;S0880SSSSSX2880X @
,X 7S82 8ZZZS7XX7XX77SrXXX777SZZZZ XZar i;
:. r02 .,.,iii.,i,,..,,. .,,,..,ii,.iii, MaX ,
8 7S; .ZZZZZZZZ8aZZZZZZ: rS Z
M :a288ZX ;XS080SS0X; iX82S0882Xr XS08a2X M
2; a888880 .; 0888880 M
B . M
B : B
B .i M
Xr . @
M .i M
a. . B
M @
Z .; 7;
XX . iZ
7X : rS
r2 .i r2
.8 . Sr
0. .i W
iW . Br
22 Sa
@X .; 7M
aZ: . 8a
iZa : 2a7
228S .i 70X2
888X0i ,0S288
iX0SXZ88ZBMZ888XX0X;

**
Polar equation: ***r*^{2} = *b*^{2} - ( *a*sin(Θ) ± √(*c*^{2} - *a*^{2}cos^{2}(Θ)) )^{2}

Cartesian equation: 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}

The curve is named after James Watt, the very same Scottish engineer who developed the steam engine. In fact the curve comes from the related problem of linkages of rods connecting two wheels of equal diameter.

Let two wheels of radius *b* have their centers 2*a* apart. Suppose that a rod of length 2*c* is fixed at each end to the circumference of the two wheels. Let *P* be the mid-point of the rod. Then Watt's Curve *C* is the locus of *P*.

If *a* = *c* then *C* is a circle of radius *b* with a figure 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 by a linkage in this way.