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LEMNISCATE OF BERNOULLI
Curve studied by Jacques Bernoulli in 1694 and Fagnano
in 1750.
Jacques Bernoulli (1654 1705): Swiss mathematician. 
Bipolar equation: (where d is the halfdistance between the poles F and F', foci of the lemniscate). Tripolar equation: (O middle of F and F'). Polar equation: (with , F(d, 0), F'(d,0)). 
Cartesian equation: .
Rational bicircular quartic. Cartesian parametrization: (). Rational Cartesian parametrization: (, ), hence the complex parametrization: . Another Cartesian parametrization: (). Cartesian parametrization of the complexified curve: . Complex equation: . Pedal equation: . Polar tangential angle: . Curvilinear abscissa: . Radius of curvature: . Intrinsic equation:. Length: , where , elliptic integral of the first kind, is the constant of the lemniscate (variation of the letter "pi"), to relate to . We also have: 
The lemniscate of Bernoulli is a challenger, along with the cardioid, for the record number of memberships to various families of remarkable curves.
It is indeed:
 a special case of Cassinian oval (see the bipolar equation)  
 a special case of Booth curve.  
 a special case of sinusoidal spiral (see the polar equation)  
 as all rational bicircular quartic, at the same time, the pedal with respect to O and the inverse (reference circle with diameter [A(a,0) ; A'(a,0)]) of the rectangular hyperbola with centre O and vertices A and A'; F and F' are the inverses of the foci of this hyperbola and the tangents at the origin are the inverses of the asymptotes.  
 it is also, as a pedal curve, the envelope of the circles with diameters the ends of which are the centre and a point of this hyperbola.  
 as well as the locus of the centre of a hyperbola rolling without slipping on an equal hyperbola, with coinciding vertices. 

 the cissoid of the circle with centre F passing by O and the circle with centre and radius a.  
 the cissoid with pole O of the circles (C)
and (C') with centres F and F' and radii
a/2.
In dotted lines, the circles (C) and (C'), in blue their homothetic image, the median of which is the lemniscate. 

 the locus of the middles of segment lines of length 2d the ends of which describe two circles with radius a centred on F and F'. 

Therefore, the lemniscate is a curve of the threebar, in the special case of the Watt curve; according to the principle of the slidercrank exchange, there exists a second construction with an articulated quadrilateral: 

 The section of a torus, with revolution radius d and meridian radius d/2, by a plane located at distance d/2 from the axis (the lemniscate is therefore a spiric of Perseus) 

 the curve passing by O the curvature of which is proportional to the distance to O (compare to the elastic curve, the curvature of which is proportional to the distance to a fixed line)  
 the locus of the points M such that  
 the projection on the plane xOy of the biquadratic: , intersection of a cone of revolution by a paraboloid of revolution: 

Furthermore:
 the asymptotic curves of the Plücker conoid are projected on lemniscates of Bernoulli.
 the lemniscate of Bernoulli is a synodal curve of all the intersecting lines passing by the double point:
The evolute
of the lemniscate of Bernoulli is parametrized by .
Notice that the two vertices correspond to maxima of the curvature... 

...as opposed to the lemniscate of Gerono, where they correspond to minima... 

Generalisation: the pedal of the rectangular hyperbola with respect to a point on the symmetry axis is a distorted lemniscate, parametrized by . 

Watt mechanism to construct the lemniscate 

See here how to "thicken" a lemniscate:.
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© Robert FERRÉOL 2017