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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