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WATT CURVE
Curve studied by Watt in 1784.
James Watt (1736  1819): Scottish mechanic and engineer (the one in kilowatts...). Other names: lemniscoid, curve with long inflection. 
Polar parametrization:
where ;
we have (see the figure below). Polar equation: . Polar equation when a = c : . Cartesian equation of
{0}: .

A Watt curve is the locus of the middle of the connecting rod [PQ] of an articulated quadrilateral (APQB) satisfying AP = BQ, A and B being fixed  it is therefore a special case of threebar linkage curve where the cranks have equal lengths; here, A(0, a), B(0, a), AP = BQ = b, PQ =2c.
In other words, a Watt curve is the locus of the middle of a segment line with constant length joining two circles with same radii (the circle (C_{A}) and (C_{B}) with centres A and B and radii b), think of it, for example, as the middle of a rod connecting two wheels.
The curve is not empty iff
and it passes by O iff a, b, c are the lengths
of the sides of a triangle, i.e. ,
i.e. ;
then, the point O is, at the same time, double point and inflection
point of each branch, which are almost linear in a neighbourhood of O.
If we look for the points where the tangents at O
recut the curves, we find that these points coincide with O (in
other words, the contact is of order 2) when .
Watt, who was trying to obtain a motion as linear as possible for his steam engine, designed his mechanism in this case, which is the most interesting one when the two bars AP and BQ are horizontal when M passes by O, and the bar QP is vertical.
Watt's mechanism: a = 3, b = 4, c= 5.
The interest of obtaining a linear motion by an articulated
system is to avoid the friction due to the sliders, that imply greater
wear.
When the length of the connecting rod [PQ] is equal
to that of the fixed bar [AB] (c = a), the curve is
composed of the reunion of the circle (O, b) (case where
(APQB) is a parallelogram), and a Booth
curve; we get an oval if b > 2a, two circles if b
= 2a (case where the quadrilateral is a square), a lemniscate if
b
< 2a, which is a lemniscate of Bernoulli when .
NB: in the animations below, P travels along (C_{B})
at constant speed, which creates jolts in the motion of Q on (C_{B}).
case a = c < b/2: Booth oval + circle 
case a = c = b/2: three circles; for the two red small circles, the articulated quadrilateral is completely folded on itself 
case a = c > b/2: lemniscate of Booth + circle 
In a general fashion, the passage by O is obtained for , formula that also gives the polar angles of the corresponding tangents. Note that if d = 0, i.e. , then the tangents coincide.
In the case a ¹c,
the curves therefore assume the following shapes:
b > a + c et c < 2a: curve with two components 
2a £ c < b
 a:

ab < c < a: vertical eight 
b/2 £ a < c < Ö(a^{2} + b^{2}): curve with three crossings. 
c = Ö(a^{2} + b^{2}): vertical eight with tangent crossing. 
Ö(a^{2} + b^{2}) < c < a + b: vertical eight. 
In the case where the curve has two connected component, the 2 intersection points with the median symmetry axis have zero curvature if ; in the case a = 2 , c = 1 we have b = ; the approximation with a line is then far better than with the Watt mechanism, but the bars intersect, which is problematic for an industrial use.
Chebyshev linkage a = 2 , b = 5, c = 1 
Saw using the Chebyshev linkage 
But here is what becomes of the pseudolinear part when the ordinates are elongated:
See also the Roberts mechanism at threebar
linkage curve.
A mathematically linear motion obtained by an articulated
mechanism requires at least 5 moving bars, and it obtained, for example,
with the Hart
mechanism, or the Peaucellier
linkage.
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© Robert FERRÉOL 2017