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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.

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© Robert FERRÉOL2017