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ALAIN'S CURVE
Curve whose study was proposed by Alain Juhel. 
Cartesian equation: .
Rational quartic. Polar equation: . For a > b, polar equation: with and . For 0 < a < b, polar equation: with and . 
Alain's curve is the curve defined by the cartesian
equation above.
Here are its different looks:


Union of both hyperbolas . 



Hyperbola and its asymptotes. 
Cylindrical lemniscate 
Alain's curve is the projection onto the plane xOy of the intersection of the elliptical cone with the hyperbolic paraboloid . 
If a < b, the curve is also the planar projection
of the intersection
of two cylinders of revolution
with common tangent plane, the projection plane being this common plane. More exactly, the projection on the plan xOy of the cylinders with is the curve of equation . . 

See also Booth's curves,
images of the previous ones by a complex affinity.
Hereafter, animation of Alain's curves
in red, with the Booth's lemniscates
in green.
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© Robert FERRÉOL 2016