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DRAWBRIDGE CURVE
Curve defined by Bernard
Forest de Bélidor during his study of the drawbridge mechanism
that since bears his name.
He had himself named this mechanism: drawbridge with a sinusoid, even though the curve is not a sinusoid. 
The drawbridge curve is the curve described by the end of the counterweight of a drawbridge (figure above) so that the system bridge + counterweight is always in equilibrium.
It can be proved that this curve is none other than a portion of Cartesian oval.
Here is a proof of this fact:
With the notations of the opposite figure, we write that
the total potential energy is constant: ;
by eliminating a between this relation
and the AlKashi relation: ,
we get ,
which is indeed the equation of a Cartesian
oval, that reduces to the limaçon
of Pascal
if . The
point C is a focus of the oval.
If we consider the case where the counterweight is at C when the bridge is down, then the constant E equals 0 and ; the drawbridge curve is still the limaçon of Pascal , which is a cardioid if . In the general case, if and when the drawbridge is down, then and the oval is a limaçon if or . 
a = length of the drawbridge = AB

Calculation of the tension of the hoist: since the bridge is in equilibrium, the sum of the moments of the forces applied to it is zero: ; we get . 
A drawbridge with the Bélidor system can be found in Fort l'Ecluse (not far from Geneva). 
Other curves defined mechanically: the
curve
of the water bucket, the curve of
the tightrope walker.
Case where the curve of the drawbridge is a portion of a limaçon of Pascal 
Case where the drawbridge curve is a portion of a Cartesian oval. 
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© Robert FERRÉOL 2017