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MULTICARDIOOD
Curve proposed in a special case by Brocard in 1894, called "Brocard trefoil" by G. de Longchamps.
Homemade name. 
Polar equation: obtained by replacing x by and y by in the equation of the cardioid. 
The multicardioids are the images of the cardioid by the Brocard transformation.
Some examples with :
n = 2  n = 3  n = 5  n = 3/2  
=3a 




=2a 

(Brocard trefoil) 


=3/2a 




= a 




=a/2 




=0 




= – a 








=2a , n = 4/3  =3/2 a , n = 4/3  = – a, n = 3/4  =3a, n = 3/4 
The original article by Brocard, Nouvelles annales 1894, page 58
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© Robert FERRÉOL
2017