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BEETLE CURVE

Curve found under this name in 1857 in the textbook of Ecole polytechnique applicants, redacted by Catalan, and then in the treatise on analysis of Laurent.

 
Polar equation: .
Cartesian equation: .
Polar equation in the case a = b, in a frame turned by p/2: .
Rational sextic.

The beetle curves are the pedals of astroids; here, the point O is the pole of the pedal, and the centre of the astroid is A(a, b), and its parametrization is .
The polar equation shows that the beetle curves are the cissoids of the circle and the quatrefoil; and, besides, we get this trefoil when a = b = 0.

The beetle curves are to the astroid what the folia are to the deltoid.

see also perso.wanadoo.fr/nvogel/Dossiers/cadreprom6.html
 
 

case 


 case 

case a = 0, b = r, equation 


 
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© Robert FERRÉOL 2017